Ordinal analysis - Wikipedia

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id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Ordinal_notations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordinal_notations"> <div class="vector-toc-text"> 2.1 Ordinal notations </div> </a> <ul id="toc-Ordinal_notations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Upper_bound" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Upper_bound"> <div class="vector-toc-text"> 3 Upper bound </div> </a> <ul id="toc-Upper_bound-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 4 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Theories_with_proof-theoretic_ordinal_ω" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_ω"> <div class="vector-toc-text"> 4.1 Theories with proof-theoretic ordinal ω </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_ω-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_proof-theoretic_ordinal_ω2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_ω2"> <div class="vector-toc-text"> 4.2 Theories with proof-theoretic ordinal ω2 </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_ω2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_proof-theoretic_ordinal_ω3" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_ω3"> <div class="vector-toc-text"> 4.3 Theories with proof-theoretic ordinal ω3 </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_ω3-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_proof-theoretic_ordinal_ωn_(for_n_=_2,_3,_..._ω)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_ωn_(for_n_=_2,_3,_..._ω)"> <div class="vector-toc-text"> 4.4 Theories with proof-theoretic ordinal ωn (for n = 2, 3, ... ω) </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_ωn_(for_n_=_2,_3,_..._ω)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_proof-theoretic_ordinal_ωω" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_ωω"> <div class="vector-toc-text"> 4.5 Theories with proof-theoretic ordinal ωω </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_ωω-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_proof-theoretic_ordinal_ε0" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_ε0"> <div class="vector-toc-text"> 4.6 Theories with proof-theoretic ordinal ε0 </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_ε0-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_proof-theoretic_ordinal_the_Feferman–Schütte_ordinal_Γ0" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_the_Feferman–Schütte_ordinal_Γ0"> <div class="vector-toc-text"> 4.7 Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ0 </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_the_Feferman–Schütte_ordinal_Γ0-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_proof-theoretic_ordinal_the_Bachmann–Howard_ordinal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_proof-theoretic_ordinal_the_Bachmann–Howard_ordinal"> <div class="vector-toc-text"> 4.8 Theories with proof-theoretic ordinal the Bachmann–Howard ordinal </div> </a> <ul id="toc-Theories_with_proof-theoretic_ordinal_the_Bachmann–Howard_ordinal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_with_larger_proof-theoretic_ordinals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theories_with_larger_proof-theoretic_ordinals"> <div class="vector-toc-text"> 4.9 Theories with larger proof-theoretic ordinals </div> </a> <ul id="toc-Theories_with_larger_proof-theoretic_ordinals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Table_of_ordinal_analyses" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Table_of_ordinal_analyses"> <div class="vector-toc-text"> 5 Table of ordinal analyses </div> </a> <button aria-controls="toc-Table_of_ordinal_analyses-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Table of ordinal analyses subsection </button> <ul id="toc-Table_of_ordinal_analyses-sublist" class="vector-toc-list"> <li id="toc-Key" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Key"> <div class="vector-toc-text"> 5.1 Key </div> </a> <ul id="toc-Key-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 6 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 7 Notes </div> </a> <ul id="toc-Notes-sublist" 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0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations. (September 2021) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical technique used in proof theory</div> In <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, ordinal analysis assigns <a href="/wiki/Ordinal_number" title="Ordinal number">ordinals</a> (often <a href="/wiki/Large_countable_ordinals" class="mw-redirect" title="Large countable ordinals">large countable ordinals</a>) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often <a href="/wiki/Equiconsistency" title="Equiconsistency">equiconsistent</a>, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bb95191e7af7bd30330e983e82620aa5b2a1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}}" /> functions of the theory.<a href="#cite_note-1">[1]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=1" title="Edit section: History">edit</a>]</div> The field of ordinal analysis was formed when <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> in 1934 used <a href="/wiki/Cut_elimination" class="mw-redirect" title="Cut elimination">cut elimination</a> to prove, in modern terms, that the proof-theoretic ordinal of <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> is <a href="/wiki/Epsilon_numbers_(mathematics)" class="mw-redirect" title="Epsilon numbers (mathematics)">ε0</a>. See <a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">Gentzen's consistency proof</a>. <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof-theoretic ordinal of such a theory <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /> is the supremum of the <a href="/wiki/Order_type" title="Order type">order types</a> of all <a href="/wiki/Ordinal_notation" title="Ordinal notation">ordinal notations</a> (necessarily <a href="/wiki/Recursive_ordinal" class="mw-redirect" title="Recursive ordinal">recursive</a>, see next section) that the theory can prove are <a href="/wiki/Well-founded_relation" title="Well-founded relation">well founded</a>—the supremum of all ordinals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> for which there exists a <a href="/wiki/Kleene%27s_O" title="Kleene's O">notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}" /> in Kleene's sense</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /> proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}" /> is an ordinal notation. Equivalently, it is the supremum of all ordinals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> such that there exists a <a href="/wiki/Computable_function" title="Computable function">recursive relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /> (the set of natural numbers) that <a href="/wiki/Well-order" title="Well-order">well-orders</a> it with ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> and such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /> proves <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a> of arithmetical statements for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />. <div class="mw-heading mw-heading3"><h3 id="Ordinal_notations">Ordinal notations</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=3" title="Edit section: Ordinal notations">edit</a>]</div> Some theories, such as subsystems of second-order arithmetic, have no conceptualization or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /> of Z2 to "prove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> well-ordered", we instead construct an <a href="/wiki/Ordinal_notation" title="Ordinal notation">ordinal notation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,{\tilde {<}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo><</mo> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,{\tilde {<}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57462c00fa7b099b83a3916ab17c34472c786000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,{\tilde {<}})}" /> with order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" />. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /> can now work with various transfinite induction principles along <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,{\tilde {<}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo><</mo> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,{\tilde {<}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57462c00fa7b099b83a3916ab17c34472c786000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,{\tilde {<}})}" />, which substitute for reasoning about set-theoretic ordinals. However, some pathological notation systems exist that are unexpectedly difficult to work with. For example, Rathjen gives a primitive recursive notation system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ,<_{T})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <msub> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ,<_{T})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da65d77849ffd4086dc89eb6a6781e3479dc29f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.719ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ,<_{T})}" /> that is well-founded iff PA is consistent,<a href="#cite_note-Realm-2">[2]</a>p. 3 despite having order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /> - including such a notation in the ordinal analysis of PA would result in the false equality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PTO(PA)}}=\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">O</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> <mo>=</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PTO(PA)}}=\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060a0cd7974e81f0d42807d45a5a1e03f6fa365d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.168ex; height:2.843ex;" alt="{\displaystyle {\mathsf {PTO(PA)}}=\omega }" />. <div class="mw-heading mw-heading2"><h2 id="Upper_bound">Upper bound</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=4" title="Edit section: Upper bound">edit</a>]</div> Since an ordinal notation must be recursive, the proof-theoretic ordinal of any theory is less than or equal to the <a href="/wiki/Church%E2%80%93Kleene_ordinal" class="mw-redirect" title="Church–Kleene ordinal">Church–Kleene ordinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}^{\mathrm {CK} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">K</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}^{\mathrm {CK} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0287c1265a806f47a4187ef41cec17113c53f204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.143ex; height:3.176ex;" alt="{\displaystyle \omega _{1}^{\mathrm {CK} }}" />. In particular, the proof-theoretic ordinal of an <a href="/wiki/Inconsistent" class="mw-redirect" title="Inconsistent">inconsistent</a> theory is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}^{\mathrm {CK} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">K</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}^{\mathrm {CK} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0287c1265a806f47a4187ef41cec17113c53f204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.143ex; height:3.176ex;" alt="{\displaystyle \omega _{1}^{\mathrm {CK} }}" />, because an inconsistent theory trivially proves that all ordinal notations are well-founded. For any theory that's both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253bf5a8d60183537e1b309e490ed3cd7583f849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}}" />-axiomatizable and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd076fb78a7a5ee340920a3d846ddf67455f941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}}" />-sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253bf5a8d60183537e1b309e490ed3cd7583f849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}}" /> bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd076fb78a7a5ee340920a3d846ddf67455f941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}}" />-soundness. Thus the proof-theoretic ordinal of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd076fb78a7a5ee340920a3d846ddf67455f941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}}" />-sound theory that has a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253bf5a8d60183537e1b309e490ed3cd7583f849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.732ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}}" /> axiomatization will always be a (countable) <a href="/wiki/Recursive_ordinal" class="mw-redirect" title="Recursive ordinal">recursive ordinal</a>, that is, strictly less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}^{\mathrm {CK} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">K</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}^{\mathrm {CK} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0287c1265a806f47a4187ef41cec17113c53f204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.143ex; height:3.176ex;" alt="{\displaystyle \omega _{1}^{\mathrm {CK} }}" />. <a href="#cite_note-Realm-2">[2]</a>Theorem 2.21 <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=5" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_ω">Theories with proof-theoretic ordinal ω</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=6" title="Edit section: Theories with proof-theoretic ordinal ω">edit</a>]</div> <ul><li>Q, <a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a> (although the definition of the proof-theoretic ordinal for such weak theories has to be tweaked)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>].</li> <li>PA–, the first-order theory of the nonnegative part of a discretely ordered ring.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_ω2">Theories with proof-theoretic ordinal ω2</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=7" title="Edit section: Theories with proof-theoretic ordinal ω2">edit</a>]</div> <ul><li>RFA, <a href="/wiki/Rudimentary_function" class="mw-redirect" title="Rudimentary function">rudimentary function</a> arithmetic.<a href="#cite_note-Krajicek-3">[3]</a></li> <li>IΔ0, arithmetic with induction on Δ0-predicates without any axiom asserting that exponentiation is total.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_ω3">Theories with proof-theoretic ordinal ω3</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=8" title="Edit section: Theories with proof-theoretic ordinal ω3">edit</a>]</div> <ul><li>EFA, <a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function arithmetic</a>.</li> <li>IΔ0 + exp, arithmetic with induction on Δ0-predicates augmented by an axiom asserting that exponentiation is total.</li> <li>RCA* 0, a second order form of EFA sometimes used in <a href="/wiki/Reverse_mathematics" title="Reverse mathematics">reverse mathematics</a>.</li> <li>WKL* 0, a second order form of EFA sometimes used in <a href="/wiki/Reverse_mathematics" title="Reverse mathematics">reverse mathematics</a>.</li></ul> Friedman's <a href="/wiki/Grand_conjecture" class="mw-redirect" title="Grand conjecture">grand conjecture</a> suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal. <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_ωn_(for_n_=_2,_3,_..._ω)">Theories with proof-theoretic ordinal ωn (for n = 2, 3, ... ω)</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=9" title="Edit section: Theories with proof-theoretic ordinal ωn (for n = 2, 3, ... ω)">edit</a>]</div> <ul><li>IΔ0 or EFA augmented by an axiom ensuring that each element of the n-th level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f57d70ee7aeb520dba041fbb5edd2bbce74378cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.555ex; height:2.343ex;" alt="{\displaystyle {\mathcal {E}}^{n}}" /> of the <a href="/wiki/Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a> is total.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_ωω">Theories with proof-theoretic ordinal ωω</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=10" title="Edit section: Theories with proof-theoretic ordinal ωω">edit</a>]</div> <ul><li>RCA0, <a href="/wiki/Reverse_mathematics#The_base_system_RCA0" title="Reverse mathematics">recursive comprehension</a>.</li> <li>WKL0, <a href="/wiki/Reverse_mathematics#Weak_Kőnig's_lemma_WKL0" title="Reverse mathematics">weak Kőnig's lemma</a>.</li> <li>PRA, <a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive arithmetic</a>.</li> <li>IΣ1, arithmetic with induction on Σ1-predicates.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_ε0">Theories with proof-theoretic ordinal ε0</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=11" title="Edit section: Theories with proof-theoretic ordinal ε0">edit</a>]</div> <ul><li>PA, <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> (<a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">shown</a> by <a href="/wiki/Gentzen" class="mw-redirect" title="Gentzen">Gentzen</a> using <a href="/wiki/Cut_elimination" class="mw-redirect" title="Cut elimination">cut elimination</a>).</li> <li>ACA0, <a href="/wiki/Reverse_mathematics#Arithmetical_comprehension_ACA0" title="Reverse mathematics">arithmetical comprehension</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_the_Feferman–Schütte_ordinal_Γ0">Theories with proof-theoretic ordinal the <a href="/wiki/Feferman%E2%80%93Sch%C3%BCtte_ordinal" title="Feferman–Schütte ordinal">Feferman–Schütte ordinal Γ0</a></h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=12" title="Edit section: Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ0">edit</a>]</div> <ul><li>ATR0, <a href="/wiki/Arithmetical_transfinite_recursion" class="mw-redirect" title="Arithmetical transfinite recursion">arithmetical transfinite recursion</a>.</li> <li><a href="/wiki/Martin-L%C3%B6f_type_theory" class="mw-redirect" title="Martin-Löf type theory">Martin-Löf type theory</a> with arbitrarily many finite level universes.</li></ul> This ordinal is sometimes considered to be the upper limit for "predicative" theories. <div class="mw-heading mw-heading3"><h3 id="Theories_with_proof-theoretic_ordinal_the_Bachmann–Howard_ordinal">Theories with proof-theoretic ordinal the <a href="/wiki/Bachmann%E2%80%93Howard_ordinal" title="Bachmann–Howard ordinal">Bachmann–Howard ordinal</a></h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=13" title="Edit section: Theories with proof-theoretic ordinal the Bachmann–Howard ordinal">edit</a>]</div> <ul><li>ID1, the first <a href="/wiki/Buchholz%27s_ID_hierarchy" class="mw-redirect" title="Buchholz's ID hierarchy">theory of inductive definitions</a>.</li> <li>KP, <a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek set theory</a> with the <a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">axiom of infinity</a>.</li> <li>CZF, Aczel's <a href="/wiki/CZF" class="mw-redirect" title="CZF">constructive Zermelo–Fraenkel set theory</a>.</li> <li>EON, a weak variant of the <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Feferman</a>'s explicit mathematics system T0.</li></ul> The Kripke-Platek or CZF set theories are weak set theories without axioms for the full powerset given as set of all subsets. Instead, they tend to either have axioms of restricted separation and formation of new sets, or they grant existence of certain function spaces (exponentiation) instead of carving them out from bigger relations. <div class="mw-heading mw-heading3"><h3 id="Theories_with_larger_proof-theoretic_ordinals">Theories with larger proof-theoretic ordinals</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=14" title="Edit section: Theories with larger proof-theoretic ordinals">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div>Unsolved problem in mathematics:</div> <div class="unsolved-body">What is the proof-theoretic ordinal of full second-order arithmetic?<a href="#cite_note-RathjenFromArithmetic-4">[4]</a></div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mbox{-}}{\mathsf {CA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-</mtext> </mstyle> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mbox{-}}{\mathsf {CA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5384df1452cb111a99d0f3ef126874c740a1940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.662ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mbox{-}}{\mathsf {CA}}_{0}}" />, <a href="/wiki/Second_order_arithmetic" class="mw-redirect" title="Second order arithmetic">Π11 comprehension</a> has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams",<a href="#cite_note-5">[5]</a>p. 13 and which is bounded by <a href="/wiki/Buchholz%27s_ordinal" title="Buchholz's ordinal">ψ0(Ωω)</a> in <a href="/wiki/Buchholz%27s_notation" class="mw-redirect" title="Buchholz's notation">Buchholz's notation</a>. It is also the ordinal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ID_{<\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ID_{<\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad97f7f1c537fec597ce09f5770635cb9e23c8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.629ex; height:2.509ex;" alt="{\displaystyle ID_{<\omega }}" />, the theory of finitely iterated inductive definitions. And also the ordinal of MLW, Martin-Löf type theory with indexed W-Types <a href="#CITEREFSetzer2004">Setzer (2004)</a>.</li> <li>IDω, the <a href="/wiki/Buchholz%27s_ID_hierarchy" class="mw-redirect" title="Buchholz's ID hierarchy">theory of ω-iterated inductive definitions</a>. Its proof-theoretic ordinal is equal to the <a href="/wiki/Takeuti%E2%80%93Feferman%E2%80%93Buchholz_ordinal" title="Takeuti–Feferman–Buchholz ordinal">Takeuti-Feferman-Buchholz ordinal</a>.</li> <li>T0, Feferman's constructive system of explicit mathematics has a larger proof-theoretic ordinal, which is also the proof-theoretic ordinal of the KPi, Kripke–Platek set theory with iterated admissibles and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{2}^{1}{\mbox{-}}{\mathsf {AC}}+{\mathsf {BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{2}^{1}{\mbox{-}}{\mathsf {AC}}+{\mathsf {BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe04a3daff2d80b0ffcc04058abdd3d6b4893f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.58ex; height:3.176ex;" alt="{\displaystyle \Sigma _{2}^{1}{\mbox{-}}{\mathsf {AC}}+{\mathsf {BI}}}" />.</li> <li>KPi, an extension of <a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek set theory</a> based on a <a href="/wiki/Admissible_ordinal" title="Admissible ordinal">recursively inaccessible ordinal</a>, has a very large proof-theoretic ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\varepsilon _{I+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\varepsilon _{I+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514cef023ca0d237f8161bcaee5999f26efc091a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.567ex; height:2.843ex;" alt="{\displaystyle \psi (\varepsilon _{I+1})}" /> described in a 1983 paper of Jäger and Pohlers, where I is the smallest inaccessible.<a href="#cite_note-6">[6]</a> This ordinal is also the proof-theoretic ordinal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}{\mbox{-}}{\mathsf {CA}}+{\mathsf {BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}{\mbox{-}}{\mathsf {CA}}+{\mathsf {BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09acee43dc510217004a2c499829c15f85c65440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.838ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}{\mbox{-}}{\mathsf {CA}}+{\mathsf {BI}}}" />.</li> <li>KPM, an extension of <a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek set theory</a> based on a <a href="/wiki/Admissible_ordinal" title="Admissible ordinal">recursively Mahlo ordinal</a>, has a very large proof-theoretic ordinal θ, which was described by <a href="#CITEREFRathjen1990">Rathjen (1990)</a>.</li> <li>TTM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\Omega _{1}}(\Omega _{M+\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>+</mo> <mi>ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\Omega _{1}}(\Omega _{M+\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eca0bec00401ec8d840f98e61a79dd6211942df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.511ex; height:3.009ex;" alt="{\displaystyle \psi _{\Omega _{1}}(\Omega _{M+\omega })}" />.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}+\Pi _{3}-Ref}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mo>+</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>R</mi> <mi>e</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}+\Pi _{3}-Ref}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5e377e16ef4871c3589120ecf04ffcedd9f2fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.703ex; height:2.509ex;" alt="{\displaystyle {\mathsf {KP}}+\Pi _{3}-Ref}" /> has a proof-theoretic ordinal equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\varepsilon _{K+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\varepsilon _{K+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516f707e5e057fb296c11abf3d2b77553ed1e9e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.495ex; height:2.843ex;" alt="{\displaystyle \Psi (\varepsilon _{K+1})}" />, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /> refers to the first weakly compact, due to (Rathjen 1993)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}+\Pi _{\omega }-Ref}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mo>+</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>−</mo> <mi>R</mi> <mi>e</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}+\Pi _{\omega }-Ref}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b540988c130e92ca53f94bb9e99584cf370e503f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.903ex; height:2.509ex;" alt="{\displaystyle {\mathsf {KP}}+\Pi _{\omega }-Ref}" /> has a proof-theoretic ordinal equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{X}^{\varepsilon _{\Xi +1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ξ</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{X}^{\varepsilon _{\Xi +1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c871fe56c605bbe418707893e52e222eb71540bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.566ex; height:3.176ex;" alt="{\displaystyle \Psi _{X}^{\varepsilon _{\Xi +1}}}" />, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fcfdbcd1348cf9e34618a31dbdcb36361406220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \Xi }" /> refers to the first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{0}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{0}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2585e973168fcf1f1ab9ec70ab0e8887d6bf6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{0}^{2}}" />-indescribable and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {X} =(\omega ^{+};P_{0};\epsilon ,\epsilon ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">X</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>;</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>;</mo> <mi>ϵ</mi> <mo>,</mo> <mi>ϵ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {X} =(\omega ^{+};P_{0};\epsilon ,\epsilon ,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ead65fdd0de89d3a87fbdb770b27145e2663ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.275ex; height:3.009ex;" alt="{\displaystyle \mathbb {X} =(\omega ^{+};P_{0};\epsilon ,\epsilon ,0)}" />, due to (Stegert 2010).</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Stability}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">b</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">y</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Stability}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ad48acf3b8e5c9d3596e36eb7b588569eae57e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.033ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Stability}}}" /> has a proof-theoretic ordinal equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\mathbb {X} }^{\varepsilon _{\Upsilon +1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">X</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Υ</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{\mathbb {X} }^{\varepsilon _{\Upsilon +1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9757acebe9c63a67da0ed3a0f5686e0586b51831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.714ex; height:3.343ex;" alt="{\displaystyle \Psi _{\mathbb {X} }^{\varepsilon _{\Upsilon +1}}}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Upsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Υ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Upsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e8c0694e5270ebf60de3a794a27f94a063020a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Upsilon }" /> is a cardinal analogue of the least ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b80a3fdecb9cf75091789bb4335a1bb3561b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.66ex; height:2.509ex;" alt="{\displaystyle \alpha +\beta }" />-stable for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta <\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo><</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta <\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a2847a928d4dc604cb25f967a3911fa4cc3f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \beta <\alpha }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {X} =(\omega ^{+};P_{0};\epsilon ,\epsilon ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">X</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>;</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>;</mo> <mi>ϵ</mi> <mo>,</mo> <mi>ϵ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {X} =(\omega ^{+};P_{0};\epsilon ,\epsilon ,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ead65fdd0de89d3a87fbdb770b27145e2663ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.275ex; height:3.009ex;" alt="{\displaystyle \mathbb {X} =(\omega ^{+};P_{0};\epsilon ,\epsilon ,0)}" />, due to (Stegert 2010).</li></ul> Most theories capable of describing the power set of the natural numbers have proof-theoretic ordinals that are so large that no explicit combinatorial description has yet been given. This includes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{2}^{1}-CA_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <mi>C</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{2}^{1}-CA_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6661827027d1c0266477746bf47cf15e1075290a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.201ex; height:3.176ex;" alt="{\displaystyle \Pi _{2}^{1}-CA_{0}}" />, full <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{\infty }^{1}-CA_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <mi>C</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{\infty }^{1}-CA_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f32facf424107f3ebf0c54e545e1d4038d6c012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.023ex; height:2.843ex;" alt="{\displaystyle \Pi _{\infty }^{1}-CA_{0}}" />) and set theories with powersets including <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZF</a> and ZFC. The strength of <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic</a> ZF (IZF) equals that of ZF. <div class="mw-heading mw-heading2"><h2 id="Table_of_ordinal_analyses">Table of ordinal analyses</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=15" title="Edit section: Table of ordinal analyses">edit</a>]</div> <table class="wikitable"> <caption>Table of proof-theoretic ordinals </caption> <tbody><tr> <th>Ordinal </th> <th>First-order arithmetic </th> <th>Second-order arithmetic </th> <th>Kripke-Platek set theory </th> <th>Type theory </th> <th>Constructive set theory </th> <th>Explicit mathematics </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b954e6612a5bc83e108fd4eda3c54901a2786f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.711ex; height:2.343ex;" alt="{\displaystyle {\mathsf {Q}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc60ab391d9835017f0778767fb25a54402d20f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.5ex; height:2.676ex;" alt="{\displaystyle \omega ^{2}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RFA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RFA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62fa1b178e47c519061f20c0ca3c76ba23b17f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.375ex; height:2.176ex;" alt="{\displaystyle {\mathsf {RFA}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Delta }}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Delta }}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b83607f7f952ec79ef9fd3e3fc5c939c1c406f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.637ex; height:2.509ex;" alt="{\displaystyle {\mathsf {I\Delta }}_{0}}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d0c825c8643660254ef82621e961101dc8427b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.5ex; height:2.676ex;" alt="{\displaystyle \omega ^{3}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EFA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EFA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f822e132414d381dc8bae0ec41097739a72f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.261ex; height:2.176ex;" alt="{\displaystyle {\mathsf {EFA}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4913b90578aec06daa7907274083a49f0d4063b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.094ex; height:3.176ex;" alt="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}_{0}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}_{0}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/387dd1e66e5c4f3d18d3cbc2c904866b74837c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.843ex;" alt="{\displaystyle {\mathsf {RCA}}_{0}^{*}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {WKL}}_{0}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {WKL}}_{0}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7da1ced71d326bc9f74da8ee40b7aab8f7652c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.121ex; height:2.843ex;" alt="{\displaystyle {\mathsf {WKL}}_{0}^{*}}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbdd97cf8db5a2f29f648ec7c86cb441d89bfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.664ex; height:2.343ex;" alt="{\displaystyle \omega ^{n}}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_a">[1]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EFA}}^{\mathsf {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EFA}}^{\mathsf {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05790454736255100deb5ccd8b2d197f7d053a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.343ex; height:2.343ex;" alt="{\displaystyle {\mathsf {EFA}}^{\mathsf {n}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mo mathvariant="sans-serif">+</mo> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2ea8f071f17c5881b2629d44249ff8437db781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.943ex; height:3.176ex;" alt="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa5636cabcea62c44c8b91fd9095e06054a6fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.7ex; height:2.343ex;" alt="{\displaystyle \omega ^{\omega }}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PRA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PRA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc753f99fc799cd0ee9fb4c163fd66a551e4d653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.537ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PRA}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Sigma }}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Sigma }}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3e869994d0704d1d11236dc6abb77d3f2952c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:2.509ex;" alt="{\displaystyle {\mathsf {I\Sigma }}_{1}}" /><a href="#cite_note-AvigadSommer97-7">[7]</a>p. 13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25465d37b9757defbfdec0ceae66f5281b79ae2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.591ex; height:2.509ex;" alt="{\displaystyle {\mathsf {RCA}}_{0}}" /><a href="#cite_note-AvigadSommer97-7">[7]</a>p. 13, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {WKL}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {WKL}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f2e0165d0148181304ad8f47791d2a494d7417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.121ex; height:2.509ex;" alt="{\displaystyle {\mathsf {WKL}}_{0}}" /><a href="#cite_note-AvigadSommer97-7">[7]</a>p. 13 </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CPRC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CPRC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9fc5f80a0e1166f073df28e390b969514f8cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.957ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CPRC}}}" /> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\omega ^{\omega ^{\omega }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\omega ^{\omega ^{\omega }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7056ed42a018cbb31e02ef71dfffe977622129b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.658ex; height:3.176ex;" alt="{\displaystyle \omega ^{\omega ^{\omega ^{\omega }}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I}}\Sigma _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I}}\Sigma _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99cd51d17e3e5cf2207f80444f4b27c619ca9134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:2.509ex;" alt="{\displaystyle {\mathsf {I}}\Sigma _{3}}" /><a href="#cite_note-8">[8]</a><a href="#cite_note-AvigadSommer97-7">[7]</a>p. 13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}_{0}+(\Pi _{2}^{0})^{-}{\mathsf {-IND}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}_{0}+(\Pi _{2}^{0})^{-}{\mathsf {-IND}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d79f683a8b0cc3325b720d76a555a7c3896f48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.328ex; height:3.176ex;" alt="{\displaystyle {\mathsf {RCA}}_{0}+(\Pi _{2}^{0})^{-}{\mathsf {-IND}}}" /><a href="#cite_note-9">[9]</a>: 40  </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb0a8377db20e42274444cb181d51b5532b5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6812979524cbdd07e79311aa1263ab6db58758f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PA}}}" /><a href="#cite_note-AvigadSommer97-7">[7]</a>p. 13 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c859a47f5d58a4af830b235a0852a326b9febca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.64ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ACA}}_{0}}" /><a href="#cite_note-AvigadSommer97-7">[7]</a>p. 13, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d24f7a942e4be0e5833f65ddd31871479327c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.888ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA}}_{0}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-AC}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-AC}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27eb24df68d2ad2cdf4e846d0969ed06419f674b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.63ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-AC}}_{0}}" /><a href="#cite_note-AvigadSommer97-7">[7]</a>p. 13, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{R-}}{\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>R-</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{R-}}{\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e34943ee6342e710efd6364c0b080243122b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.173ex; height:3.009ex;" alt="{\displaystyle {\text{R-}}{\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 8, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c99467d9a58b436d5b0a0b6e9bc0f4018be11aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.537ex; height:2.176ex;" alt="{\displaystyle {\mathsf {RCA}}}" /><a href="#cite_note-FSS83-11">[11]</a>p. 148, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {WKL}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {WKL}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2977470fe0a7b38c5dfdb201e17b02b8aca847f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.067ex; height:2.176ex;" alt="{\displaystyle {\mathsf {WKL}}}" /><a href="#cite_note-FSS83-11">[11]</a>p. 148 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {KPu} ^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {KPu} ^{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1d16b472ee3996ee6ddab5dc9a5b870fdc73f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.657ex; height:2.343ex;" alt="{\displaystyle \mathrm {KPu} ^{r}}" /><a href="#cite_note-Jäger84-12">[12]</a>p. 869 </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ee7cbff02ef448768cfae28673e1020c89e9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.475ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b83a04919faf583248143865d6e397f49b5d4af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.338ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{\omega }}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}_{0}+{\mathsf {iRT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}_{0}+{\mathsf {iRT}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974dcf989ef72aaa4eeda56348afbf06f212040e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.121ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ACA}}_{0}+{\mathsf {iRT}}}" />,<a href="#cite_note-13">[13]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}_{0}+\forall Y\forall n\exists X({\textrm {TJ}}(n,X,Y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">∀</mi> <mi>Y</mi> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mi mathvariant="normal">∃</mi> <mi>X</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>TJ</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}_{0}+\forall Y\forall n\exists X({\textrm {TJ}}(n,X,Y))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c8141d98e974d58fbd4ddd2d212d0d649e7ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.165ex; height:2.843ex;" alt="{\displaystyle {\mathsf {RCA}}_{0}+\forall Y\forall n\exists X({\textrm {TJ}}(n,X,Y))}" /><a href="#cite_note-MarconeMontalbán2010-14">[14]</a>: 8  </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\varepsilon _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\varepsilon _{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c49ed8d5baffdbb17121fdababad70d7b83a5e31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.913ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{\varepsilon _{0}}}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a91137e338ef377365b38f8c19d94d5dc11b618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.586ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ACA}}}" /><a href="#cite_note-15">[15]</a>p. 959 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acfae32a71eb98bc09c043232740ca8bc5b94452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.073ex; height:2.509ex;" alt="{\displaystyle \zeta _{0}}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}_{0}+\forall X\exists Y({\textrm {TJ}}(\omega ,X,Y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">∀</mi> <mi>X</mi> <mi mathvariant="normal">∃</mi> <mi>Y</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>TJ</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}_{0}+\forall X\exists Y({\textrm {TJ}}(\omega ,X,Y))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0b94a036f88b7153d53856fae68326776dc683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.578ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ACA}}_{0}+\forall X\exists Y({\textrm {TJ}}(\omega ,X,Y))}" />,<a href="#cite_note-Heissenbüttel2001-16">[16]</a><a href="#cite_note-MarconeMontalbán2010-14">[14]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {p}}_{1}({\mathsf {ACA}}_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {p}}_{1}({\mathsf {ACA}}_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9c4e2d9d6257999eafd3ae45e5413375709f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.706ex; height:2.843ex;" alt="{\displaystyle {\mathsf {p}}_{1}({\mathsf {ACA}}_{0})}" />,<a href="#cite_note-Probst2017-17">[17]</a>: 7  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RFN}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RFN}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e78b188d785c1d09fb9e4275a6e0406e093e2e27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle {\mathsf {RFN}}_{0}}" /><a href="#cite_note-Heissenbüttel2001-16">[16]</a>p. 17, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}_{0}+({\mathsf {BR}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}_{0}+({\mathsf {BR}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67d18aa1306a09d70bc2f81592be0d8d9da9043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.342ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ACA}}_{0}+({\mathsf {BR}})}" /><a href="#cite_note-Heissenbüttel2001-16">[16]</a>p. 5 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (2,\varepsilon _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (2,\varepsilon _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088169d0281112876f40cdf78960e03ef50c2c01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.664ex; height:2.843ex;" alt="{\displaystyle \varphi (2,\varepsilon _{0})}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RFN}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RFN}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ac01dcd6cb007ca2590892d5ca78cf9e60cd04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.47ex; height:2.176ex;" alt="{\displaystyle {\mathsf {RFN}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}+\forall X\exists Y({\textrm {TJ}}(\omega ,X,Y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mo>+</mo> <mi mathvariant="normal">∀</mi> <mi>X</mi> <mi mathvariant="normal">∃</mi> <mi>Y</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>TJ</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}+\forall X\exists Y({\textrm {TJ}}(\omega ,X,Y))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc8d9725a3244adbeacd319f43fca5160c7f255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.523ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ACA}}+\forall X\exists Y({\textrm {TJ}}(\omega ,X,Y))}" /><a href="#cite_note-Heissenbüttel2001-16">[16]</a>p. 52 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\omega ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\omega ,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d432f3310d2848755f079c46de77e97acae80351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.972ex; height:2.843ex;" alt="{\displaystyle \varphi (\omega ,0)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{1}\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{1}\#}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6de6b328bf1a541fd55f9051a90ba23088daf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.315ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{1}\#}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6590f8ab632875babe92caa0add6d9a9c3c2c12b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.785ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CR}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}{\mathsf {-DC}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}{\mathsf {-DC}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae8ea795c06fa65fab6066ae1788d164f034c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.758ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}{\mathsf {-DC}}_{0}}" /><a href="#cite_note-18">[18]</a> </td> <td> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+JR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">J</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+JR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87fe56509e950873e584351c442292997f9ddc40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.883ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+JR}}}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\varepsilon _{0},0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\varepsilon _{0},0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac0c33da90bea1d7de3d322e1d8c3af8f360988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.664ex; height:2.843ex;" alt="{\displaystyle \varphi (\varepsilon _{0},0)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ee50067aca8b61a1479a693d3eebf175090822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:3.389ex; height:3.343ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{1}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KFL}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KFL}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a9580aeb0b86c0cbf0ded88d3300df70a78210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KFL}}}" /><a href="#cite_note-FNF20-19">[19]</a>p. 17, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KF}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df58263e64f33e23f050f55620dc747c63c8b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.936ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KF}}}" /><a href="#cite_note-FNF20-19">[19]</a>p. 17 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db22c2358be0954f459a9a55cca7dc3d108ab438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.834ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA}}}" /><a href="#cite_note-Simpson85-20">[20]</a>p. 140, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-AC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965be4d4896a955c51ec2bdae9161004492c4187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.576ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-AC}}}" /><a href="#cite_note-Simpson85-20">[20]</a>p. 140, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e7b5a102b9636d6858db4bdbe2b1d16939e654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.704ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}}}" /><a href="#cite_note-Simpson85-20">[20]</a>p. 140, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{W-}}{\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>W-</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{W-}}{\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ca54b6d79d9b7939cdb4e4730a34447de347a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:3.009ex;" alt="{\displaystyle {\text{W-}}{\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {KPu} ^{r}+(\mathrm {IND} _{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {KPu} ^{r}+(\mathrm {IND} _{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8701ee1c5dd424cbff775f92771d9b7c6d7a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.357ex; height:2.843ex;" alt="{\displaystyle \mathrm {KPu} ^{r}+(\mathrm {IND} _{N})}" /><a href="#cite_note-Jäger84-12">[12]</a>p. 870 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8bf67c4c53b935d6689be1a242818ef4ef6b950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.348ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{1}}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">J</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98dda0d506192947f139e8e3179e39c6274e2b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.381ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J}}}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\varepsilon _{\varepsilon _{0}},0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\varepsilon _{\varepsilon _{0}},0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ad9af3242c840e4799201f4430ee01981ffe2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.439ex; height:3.009ex;" alt="{\displaystyle \varphi (\varepsilon _{\varepsilon _{0}},0)}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec02d27709ddd3ab606f7d40fd365e68a9316eba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.688ex; height:3.009ex;" alt="{\displaystyle {\widehat {\mathbf {E} {\boldsymbol {\Omega }}}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 27, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathbf {EID} }}_{\boldsymbol {1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbf {EID} }}_{\boldsymbol {1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d613814f0c3f8124a094f82236108ce7cb788ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.998ex; height:3.509ex;" alt="{\displaystyle {\widehat {\mathbf {EID} }}_{\boldsymbol {1}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 27 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\varphi (\omega ,0),0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\varphi (\omega ,0),0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d0d744f18febaef50884071214fff0ddc0682a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.497ex; height:2.843ex;" alt="{\displaystyle \varphi (\varphi (\omega ,0),0)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PRS} \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">S</mi> </mrow> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {PRS} \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7493812c716faf3751f8e0b4f4c300b65a6c30c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.032ex; height:2.176ex;" alt="{\displaystyle \mathrm {PRS} \omega }" /><a href="#cite_note-21">[21]</a>p.9 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ({\mathsf {<}}\Omega ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif"><</mo> </mrow> </mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ({\mathsf {<}}\Omega ,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0396f42e0fa390f9a782b1fd45fb90e29440ff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.012ex; height:2.843ex;" alt="{\displaystyle \varphi ({\mathsf {<}}\Omega ,0)}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_b">[2]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(ID\#)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">#</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(ID\#)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f3a50ef56904db81479a06b0038f885a878897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.662ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(ID\#)}}}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13fad944f839798f9a4fe3daec41216315a1c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:4.868ex; height:3.343ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega }}" />,<a href="#cite_note-22">[22]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {U(PA)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">U</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {U(PA)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9ed8c5ddffdf4f7c8573b1c39193b8d5481159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:2.843ex;" alt="{\displaystyle {\mathsf {U(PA)}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {KFL} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">F</mi> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {KFL} ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb16e22e9ec2fe9d989fa2f2fad1bfa88fb26d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.439ex; height:2.343ex;" alt="{\displaystyle \mathbf {KFL} ^{*}}" /><a href="#cite_note-FNF20-19">[19]</a>p. 22, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {KF} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {KF} ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9f4dcf9d02f2a82ba8be68d9f3748b0acfdf8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.831ex; height:2.343ex;" alt="{\displaystyle \mathbf {KF} ^{*}}" /><a href="#cite_note-FNF20-19">[19]</a>p. 22, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}(\mathrm {NFA} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}(\mathrm {NFA} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8aa7a1163e8a5d97f736752964ded0f4f8aafdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.038ex; width:8.448ex; height:2.843ex;" alt="{\displaystyle {\mathcal {U}}(\mathrm {NFA} )}" /><a href="#cite_note-23">[23]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee518e00cc8bc05f29a182eaf59765c1ef7624b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.689ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ATR}}_{0}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA+BR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA+BR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3e39b973bd812e31071b0be50729fc1f17ea17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.726ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{1}^{1}{\mathsf {-CA+BR}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1}^{1}\mathrm {-CA} _{0}+\mathrm {(SUB)} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">B</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1}^{1}\mathrm {-CA} _{0}+\mathrm {(SUB)} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54707db3b6cda173929f89c69b2da5a2b0a0b4d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.605ex; height:3.176ex;" alt="{\displaystyle \Delta _{1}^{1}\mathrm {-CA} _{0}+\mathrm {(SUB)} }" />,<a href="#cite_note-FJ83-24">[24]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {FP} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {FP} _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252ee78f7b58f24e8ce9d92ccb949a3b7a9c691d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.155ex; height:2.509ex;" alt="{\displaystyle \mathrm {FP} _{0}}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/437a4e04606975fbe5b63b0c4318d7f2044bfd95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.709ex; height:2.676ex;" alt="{\displaystyle {\mathsf {KPi}}^{0}}" /><a href="#cite_note-Jäger84-12">[12]</a>p. 878, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPu}}^{0}+(\mathrm {BR} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">u</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPu}}^{0}+(\mathrm {BR} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd245debdd4c57523ef2fe8973060164a84635a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.36ex; height:3.176ex;" alt="{\displaystyle {\mathsf {KPu}}^{0}+(\mathrm {BR} )}" /><a href="#cite_note-Jäger84-12">[12]</a>p. 878 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{<\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{<\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82935c3041767bfed1b2bb3c5c2a58da30d7c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.827ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{<\omega }}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {MLU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {MLU}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b814b30872af233e4e5a5656b602d181e7119b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.893ex; height:2.176ex;" alt="{\displaystyle {\mathsf {MLU}}}" /> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\omega ^{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\omega ^{\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51bbfbbe9b67c3b82d87cb7db50b2f2d5d1fd587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.702ex; height:2.509ex;" alt="{\displaystyle \Gamma _{\omega ^{\omega }}}" /> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPI}}^{0}+({\mathsf {\Sigma _{1}-I}}_{\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="sans-serif">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="sans-serif">1</mn> </mrow> </msub> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPI}}^{0}+({\mathsf {\Sigma _{1}-I}}_{\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dacf355e681c7904d9a60a5786d9b1eaad0dcde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.923ex; height:3.176ex;" alt="{\displaystyle {\mathsf {KPI}}^{0}+({\mathsf {\Sigma _{1}-I}}_{\omega })}" /><a href="#cite_note-26">[26]</a>p.13 </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\varepsilon _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\varepsilon _{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace3b7407c1d7e4a6234ad0746578ee006b31914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.283ex; height:2.843ex;" alt="{\displaystyle \Gamma _{\varepsilon _{0}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d20b45574ebd4a002281a9e3b416b54ea18ff94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:3.589ex; height:3.343ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{\omega }}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/833b7bcdd3762b701897bd2d3f2a0b718416124d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.635ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ATR}}}" /><a href="#cite_note-27">[27]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPI}}^{0}{\mathsf {+F-I}}_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">F</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPI}}^{0}{\mathsf {+F-I}}_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/033605b7f7b93bc7c5a376d6433a2fae1709a7af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.672ex; height:3.009ex;" alt="{\displaystyle {\mathsf {KPI}}^{0}{\mathsf {+F-I}}_{\omega }}" /> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,\omega ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>ω</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (1,\omega ,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14dcc1a6614d556153edf7b64ba979a692907ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.168ex; height:2.843ex;" alt="{\displaystyle \varphi (1,\omega ,0)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega ^{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega ^{\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a9f33ec7a88b32ad6e09c524e04fea266eba39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:5.862ex; height:3.343ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega ^{\omega }}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}_{0}+({\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}_{0}+({\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab829901b6536307d2323af5ae2b842dceed712b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.042ex; height:3.176ex;" alt="{\displaystyle {\mathsf {ATR}}_{0}+({\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}})}" /><a href="#cite_note-Probst2017-17">[17]</a>: 7  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}^{0}{\mathsf {+\Sigma _{1}-I}}_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <msub> <mi mathvariant="sans-serif">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="sans-serif">1</mn> </mrow> </msub> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}^{0}{\mathsf {+\Sigma _{1}-I}}_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f5e1ec3608ef35ee87aa65f6579cca911e26aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.991ex; height:3.009ex;" alt="{\displaystyle {\mathsf {KPi}}^{0}{\mathsf {+\Sigma _{1}-I}}_{\omega }}" /> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,\varepsilon _{0},0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (1,\varepsilon _{0},0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc06755500659ebc6dca654d4c6247538f611c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.843ex;" alt="{\displaystyle \varphi (1,\varepsilon _{0},0)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{<\varepsilon _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{<\varepsilon _{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca5da3e513c2077d07b19bbc5c0d32ab3dd7323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.01ex; width:5.443ex; height:3.676ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{<\varepsilon _{0}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}+({\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}+({\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ced4358fe4aaa225c12ecd9819644650cbff80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.988ex; height:3.176ex;" alt="{\displaystyle {\mathsf {ATR}}+({\mathsf {\Sigma }}_{1}^{1}{\mathsf {-DC}})}" /><a href="#cite_note-Probst2017-17">[17]</a>: 7  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}^{0}{\mathsf {+F-I}}_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">F</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}^{0}{\mathsf {+F-I}}_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d39321af7f9ac55bef4ca61cb4edf68c2ad7cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.582ex; height:3.009ex;" alt="{\displaystyle {\mathsf {KPi}}^{0}{\mathsf {+F-I}}_{\omega }}" /> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,\Gamma _{0},0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (1,\Gamma _{0},0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09810df42e9a6cf8d71633f01664ea39f49a36e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.229ex; height:2.843ex;" alt="{\displaystyle \varphi (1,\Gamma _{0},0)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{<\Gamma _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{<\Gamma _{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aae051dcb3bb9aabc3da98ecbab1cfbc0825f7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.01ex; width:5.704ex; height:3.676ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{<\Gamma _{0}}}" /> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {MLS}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {MLS}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a8c0a7c6d456061ae1ef8238669819e93cbfbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.586ex; height:2.176ex;" alt="{\displaystyle {\mathsf {MLS}}}" /> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (2,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (2,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5876032864b9eed07e966cde4730d3026c5c4496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.885ex; height:2.843ex;" alt="{\displaystyle \varphi (2,0,0)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut({\widehat {ID}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo mathvariant="sans-serif">^</mo> </mover> </mrow> </mrow> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut({\widehat {ID}})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ab492ed2e6ea464187f0b6f460d80430f26166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.726ex; height:3.509ex;" alt="{\displaystyle {\mathsf {Aut({\widehat {ID}})}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {FTR}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {FTR}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211585c9f544c4de4ca1129821538f78d83d8e96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.461ex; height:2.509ex;" alt="{\displaystyle {\mathsf {FTR}}_{0}}" /><a href="#cite_note-Strahm2000-28">[28]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax_{\Sigma _{1}^{1}{\mathsf {-AC}}}{\mathsf {TR}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax_{\Sigma _{1}^{1}{\mathsf {-AC}}}{\mathsf {TR}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f802b3b14b0a8b0e0c02271797ffe1018596725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:12.887ex; height:3.343ex;" alt="{\displaystyle Ax_{\Sigma _{1}^{1}{\mathsf {-AC}}}{\mathsf {TR}}_{0}}" /><a href="#cite_note-Ruede2002-29">[29]</a>p.1167, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax_{{\mathsf {ATR}}+\Sigma _{1}^{1}{\mathsf {-DC}}}{\mathsf {RFN}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax_{{\mathsf {ATR}}+\Sigma _{1}^{1}{\mathsf {-DC}}}{\mathsf {RFN}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cae6663662b587aadfb61a9e5637dc2cf2a0e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.918ex; height:3.343ex;" alt="{\displaystyle Ax_{{\mathsf {ATR}}+\Sigma _{1}^{1}{\mathsf {-DC}}}{\mathsf {RFN}}_{0}}" /><a href="#cite_note-Ruede2002-29">[29]</a>p.1167 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPh}}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">h</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPh}}^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/552229935ae7eaf89da21bbda3f07aa30d47aafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.355ex; height:2.676ex;" alt="{\displaystyle {\mathsf {KPh}}^{0}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(ML)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(ML)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eaadfb21c83d4b1bd3a3193cc665c086de66907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.695ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(ML)}}}" /> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (2,0,\varepsilon _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (2,0,\varepsilon _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5844a89b404ac43b6f7a302182eb4f75f445bad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.843ex;" alt="{\displaystyle \varphi (2,0,\varepsilon _{0})}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {FTR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {FTR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a86855a131e87f93c4d87b282ee837842703f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.407ex; height:2.176ex;" alt="{\displaystyle {\mathsf {FTR}}}" /><a href="#cite_note-Strahm2000-28">[28]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax_{\Sigma _{1}^{1}{\mathsf {-AC}}}{\mathsf {TR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax_{\Sigma _{1}^{1}{\mathsf {-AC}}}{\mathsf {TR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab364127ec1b79bed9ab74878793c630672331a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:11.833ex; height:3.343ex;" alt="{\displaystyle Ax_{\Sigma _{1}^{1}{\mathsf {-AC}}}{\mathsf {TR}}}" /><a href="#cite_note-Ruede2002-29">[29]</a>p.1167, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax_{{\mathsf {ATR}}+\Sigma _{1}^{1}{\mathsf {-DC}}}{\mathsf {RFN}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax_{{\mathsf {ATR}}+\Sigma _{1}^{1}{\mathsf {-DC}}}{\mathsf {RFN}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa935977ff3edf64ecc1f2efb312a0b4e08f4ad6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:17.864ex; height:3.343ex;" alt="{\displaystyle Ax_{{\mathsf {ATR}}+\Sigma _{1}^{1}{\mathsf {-DC}}}{\mathsf {RFN}}}" /><a href="#cite_note-Ruede2002-29">[29]</a>p.1167 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (2,\varepsilon _{0},0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (2,\varepsilon _{0},0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c48fc6208fb9c5453b6adc622ae21047fb0ea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.843ex;" alt="{\displaystyle \varphi (2,\varepsilon _{0},0)}" /> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPh}}_{0}+({\mathsf {F-I}}_{\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">h</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPh}}_{0}+({\mathsf {F-I}}_{\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ebbcf6bf488063af1f30e7b072dc51666bb4c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.069ex; height:2.843ex;" alt="{\displaystyle {\mathsf {KPh}}_{0}+({\mathsf {F-I}}_{\omega })}" /><a href="#cite_note-Strahm2000-28">[28]</a>: 11  </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\omega ,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\omega ,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b292f31e850eda07f42f2fd578bde4d3e1e4d4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.168ex; height:2.843ex;" alt="{\displaystyle \varphi (\omega ,0,0)}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Pi _{2}^{1}{\mathsf {-RFN}})_{0}^{\Sigma _{1}^{1}{\mathsf {-DC}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Pi _{2}^{1}{\mathsf {-RFN}})_{0}^{\Sigma _{1}^{1}{\mathsf {-DC}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e347ff7f38ac42863100cff641c51c311c58e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.65ex; height:4.176ex;" alt="{\displaystyle (\Pi _{2}^{1}{\mathsf {-RFN}})_{0}^{\Sigma _{1}^{1}{\mathsf {-DC}}}}" /><a href="#cite_note-Ruede2003-30">[30]</a>p.233, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}{\mathsf {-TDC}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}{\mathsf {-TDC}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f57f2ea9ddcfb2e368861ec573210ae12f69c762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.341ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}{\mathsf {-TDC}}_{0}}" /><a href="#cite_note-Ruede2003-30">[30]</a>p.233 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPm}}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPm}}^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798f7bbc666ec20fc990f8e40595bddf5848f731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.998ex; height:2.676ex;" alt="{\displaystyle {\mathsf {KPm}}^{0}}" /><a href="#cite_note-StrahmMahlo2002-31">[31]</a>p.276 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EMA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EMA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c00ff6ebca56d615f1ecd378cd08d1817051de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.972ex; height:2.176ex;" alt="{\displaystyle {\mathsf {EMA}}}" /><a href="#cite_note-StrahmMahlo2002-31">[31]</a>p.276 </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\varepsilon _{0},0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\varepsilon _{0},0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ab612012476bc7a2264fb31245eae1756daf64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.843ex;" alt="{\displaystyle \varphi (\varepsilon _{0},0,0)}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Pi _{2}^{1}{\mathsf {-RFN}})^{\Sigma _{1}^{1}{\mathsf {-DC}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Pi _{2}^{1}{\mathsf {-RFN}})^{\Sigma _{1}^{1}{\mathsf {-DC}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8a5cad0b52faa49a4b57dcbc70f68376f44df76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.65ex; height:3.676ex;" alt="{\displaystyle (\Pi _{2}^{1}{\mathsf {-RFN}})^{\Sigma _{1}^{1}{\mathsf {-DC}}}}" /><a href="#cite_note-Ruede2003-30">[30]</a>p.233, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}{\mathsf {-TDC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}{\mathsf {-TDC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33a71fd771ef1dfb5af25cae2a3168fd47064d9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.287ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}{\mathsf {-TDC}}}" /><a href="#cite_note-Probst2017-17">[17]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPm}}^{0}+({\mathcal {L}}^{*}{\mathsf {-I}}_{\mathsf {N}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPm}}^{0}+({\mathcal {L}}^{*}{\mathsf {-I}}_{\mathsf {N}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a331f6e499202cf6cc04c04f69e270087fcc6a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.156ex; height:3.176ex;" alt="{\displaystyle {\mathsf {KPm}}^{0}+({\mathcal {L}}^{*}{\mathsf {-I}}_{\mathsf {N}})}" /><a href="#cite_note-StrahmMahlo2002-31">[31]</a>p.277 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EMA}}+(\mathbb {L} {\mathsf {-I}}_{\mathsf {N}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EMA}}+(\mathbb {L} {\mathsf {-I}}_{\mathsf {N}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dce44f07cdbcb4ee209ff88958714c2b9d1f670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.022ex; height:2.843ex;" alt="{\displaystyle {\mathsf {EMA}}+(\mathbb {L} {\mathsf {-I}}_{\mathsf {N}})}" /><a href="#cite_note-StrahmMahlo2002-31">[31]</a>p.277 </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (1,0,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6460ed5e3f816c6c69b2476e9667cb38f65385b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.081ex; height:2.843ex;" alt="{\displaystyle \varphi (1,0,0,0)}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {p}}_{1}(\Sigma _{1}^{1}{\mathsf {-TDC}}_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {p}}_{1}(\Sigma _{1}^{1}{\mathsf {-TDC}}_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220d86a9e3479199f2362ef4c0834b806527104e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.406ex; height:3.176ex;" alt="{\displaystyle {\mathsf {p}}_{1}(\Sigma _{1}^{1}{\mathsf {-TDC}}_{0})}" /><a href="#cite_note-Probst2017-17">[17]</a>: 7  </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\Omega _{1}}(\Omega ^{\Omega ^{\omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\Omega _{1}}(\Omega ^{\Omega ^{\omega }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7707d395c063aa9eabcf7506a11b1750e79a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.664ex; height:3.509ex;" alt="{\displaystyle \psi _{\Omega _{1}}(\Omega ^{\Omega ^{\omega }})}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}_{0}^{*}+\Pi _{1}^{1}{\mathsf {-CA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}_{0}^{*}+\Pi _{1}^{1}{\mathsf {-CA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfafeeb8e85a2a7c9eb4a36c5260b2844c20680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.584ex; height:3.176ex;" alt="{\displaystyle {\mathsf {RCA}}_{0}^{*}+\Pi _{1}^{1}{\mathsf {-CA}}^{-}}" />,<a href="#cite_note-32">[32]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {p}}_{3}({\mathsf {ACA}}_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {p}}_{3}({\mathsf {ACA}}_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/850e58f0530fdfb4eb79e054b29032a2f5a94b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.706ex; height:2.843ex;" alt="{\displaystyle {\mathsf {p}}_{3}({\mathsf {ACA}}_{0})}" /><a href="#cite_note-Probst2017-17">[17]</a>: 7  </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta (\Omega ^{\Omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϑ</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta (\Omega ^{\Omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8013dc3984cc366322206385f4232d6d4e45e724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.28ex; height:3.176ex;" alt="{\displaystyle \vartheta (\Omega ^{\Omega })}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {p}}_{1}({\mathsf {p}}_{3}({\mathsf {ACA}}_{0}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {p}}_{1}({\mathsf {p}}_{3}({\mathsf {ACA}}_{0}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9e142fca827a1d0dcddebdc0656b20106cc2d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.771ex; height:2.843ex;" alt="{\displaystyle {\mathsf {p}}_{1}({\mathsf {p}}_{3}({\mathsf {ACA}}_{0}))}" /><a href="#cite_note-Probst2017-17">[17]</a>: 7  </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\varepsilon _{\Omega +1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\varepsilon _{\Omega +1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1264385f2ded93bcf20f4bd1b3c8b055779e9d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.98ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\varepsilon _{\Omega +1})}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_c">[3]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c662977612664fe5f0d8d8c8b4fd3c043d4a962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{1}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{W-}}{\widetilde {\mathbf {E} {\boldsymbol {\Omega }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>W-</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{W-}}{\widetilde {\mathbf {E} {\boldsymbol {\Omega }}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2633b796ffa88947778fc075a52a45f80e4fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.843ex;" alt="{\displaystyle {\text{W-}}{\widetilde {\mathbf {E} {\boldsymbol {\Omega }}}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94295759f4f38c35275d7f87f61ec91690ef5028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KP}}}" />,<a href="#cite_note-Realm-2">[2]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi>ω</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da18b92636475f92b90c49e811e91b0ee17d64c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.544ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KP\omega }}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {KPu} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {KPu} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c24fc68938404103d3f742032d05be502b40564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.684ex; height:2.176ex;" alt="{\displaystyle \mathrm {KPu} }" /><a href="#cite_note-Jäger84-12">[12]</a>p. 869 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{1}{\mathsf {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c99d9d51817bf51d607f93796feeaee0d16e229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.898ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {V}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68735fcc4a56810ef5e5662f4ab1a4ad4dde9e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CZF}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EON}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">O</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EON}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8de374758b8d4d7868f5387ee49c3539cc5d8ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.744ex; height:2.176ex;" alt="{\displaystyle {\mathsf {EON}}}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\varepsilon _{\Omega +\varepsilon _{0}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\varepsilon _{\Omega +\varepsilon _{0}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55e00e131ff0db9f456b6baedb326242c1f770c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.701ex; height:3.009ex;" alt="{\displaystyle \psi (\varepsilon _{\Omega +\varepsilon _{0}})}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {E} {\boldsymbol {\Omega }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {E} {\boldsymbol {\Omega }}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a29bfa5e3394d7261483fca3f3613ed90a7b557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.688ex; height:2.843ex;" alt="{\displaystyle {\widetilde {\mathbf {E} {\boldsymbol {\Omega }}}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 31, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {EID} }}_{\boldsymbol {1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mo>~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {EID} }}_{\boldsymbol {1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eea7bdcb78874112757a93488f36d1fa0530f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.998ex; height:3.343ex;" alt="{\displaystyle {\widetilde {\mathbf {EID} }}_{\boldsymbol {1}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 31, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {ACA} +(\Pi _{1}^{1}{\text{-CA}})^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>-CA</mtext> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {ACA} +(\Pi _{1}^{1}{\text{-CA}})^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9f67a0e44b9e801614192b5e16c46c9059d196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.124ex; height:3.176ex;" alt="{\displaystyle \mathbf {ACA} +(\Pi _{1}^{1}{\text{-CA}})^{-}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 31 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\varepsilon _{\Omega +\Omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\varepsilon _{\Omega +\Omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173b7055a4f2cdcadb6a4d63b16dae4812a7e7e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.29ex; height:2.843ex;" alt="{\displaystyle \psi (\varepsilon _{\Omega +\Omega })}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathsf {ID}}_{1}^{2})_{0}+{\mathsf {BR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathsf {ID}}_{1}^{2})_{0}+{\mathsf {BR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21971d6547b75a41171dea358f78cf77723b5b14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.135ex; height:3.176ex;" alt="{\displaystyle ({\mathsf {ID}}_{1}^{2})_{0}+{\mathsf {BR}}}" /><a href="#cite_note-33">[33]</a> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\varepsilon _{\varepsilon _{\Omega +1}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\varepsilon _{\varepsilon _{\Omega +1}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee69b022e0937679b3d5ac936417117c519b6f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.237ex; height:3.176ex;" alt="{\displaystyle \psi (\varepsilon _{\varepsilon _{\Omega +1}})}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} {\boldsymbol {\Omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} {\boldsymbol {\Omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc0e3b7873dcb9c2ae1d56cc6babcc248b4627c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.688ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} {\boldsymbol {\Omega }}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 33, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {EID} _{\boldsymbol {1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {EID} _{\boldsymbol {1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce0df3512e84b34dbfec9d16fe13e3443abe6fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.998ex; height:2.509ex;" alt="{\displaystyle \mathbf {EID} _{\boldsymbol {1}}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 33, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {ACA} +(\Pi _{1}^{1}{\text{-CA}})^{-}+(\mathrm {BI} _{\mathrm {PR} })^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>-CA</mtext> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {ACA} +(\Pi _{1}^{1}{\text{-CA}})^{-}+(\mathrm {BI} _{\mathrm {PR} })^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c647c3696e51955e9f09a309a0a290c13d8d846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.331ex; height:3.176ex;" alt="{\displaystyle \mathbf {ACA} +(\Pi _{1}^{1}{\text{-CA}})^{-}+(\mathrm {BI} _{\mathrm {PR} })^{-}}" /><a href="#cite_note-JagerStrahm95-10">[10]</a>p. 33 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Gamma _{\Omega +1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Gamma _{\Omega +1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93eddddc25811d5cb2eb75e526c936abe0acb9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.349ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Gamma _{\Omega +1})}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_d">[4]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {U(ID}}_{1}{\mathsf {)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">U</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {U(ID}}_{1}{\mathsf {)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13e6ef551599b70d65b9e7022f0df4eeaaa5fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.788ex; height:3.009ex;" alt="{\displaystyle {\mathsf {U(ID}}_{1}{\mathsf {)}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega }^{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega }^{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c16e48a5ffe8af797171c1ae273be6bf5fb1cc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.01ex; width:4.868ex; height:3.509ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{<\omega }^{\bullet }}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}{\mathsf {-DC}}_{0}^{\bullet }+({\mathsf {SUB}}^{\bullet })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}{\mathsf {-DC}}_{0}^{\bullet }+({\mathsf {SUB}}^{\bullet })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4256572186ae777aae6f368c00a50fa164c9c417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.904ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}{\mathsf {-DC}}_{0}^{\bullet }+({\mathsf {SUB}}^{\bullet })}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}_{0}^{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}_{0}^{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf364946c7ba0e9d8a2d5e7933d13eb4acb72f88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.689ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ATR}}_{0}^{\bullet }}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{1}^{1}{\mathsf {-AC}}_{0}^{\bullet }+({\mathsf {SUB}}^{\bullet })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}^{1}{\mathsf {-AC}}_{0}^{\bullet }+({\mathsf {SUB}}^{\bullet })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75091eb10d7bb1dc4e1ee77b50e53b6b59b9ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.776ex; height:3.176ex;" alt="{\displaystyle \Sigma _{1}^{1}{\mathsf {-AC}}_{0}^{\bullet }+({\mathsf {SUB}}^{\bullet })}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}({\mathsf {ID}}_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}({\mathsf {ID}}_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b8fd04dacb996b30ee80819303600fc1cd6d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.038ex; width:6.824ex; height:2.843ex;" alt="{\displaystyle {\mathcal {U}}({\mathsf {ID}}_{1})}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {FP}}_{0}^{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {FP}}_{0}^{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b94ddfcaf3f3abb7ee2af0910a76cb69726bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.862ex; height:2.843ex;" alt="{\displaystyle {\mathsf {FP}}_{0}^{\bullet }}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}_{0}^{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}_{0}^{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf364946c7ba0e9d8a2d5e7933d13eb4acb72f88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.689ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ATR}}_{0}^{\bullet }}" /><a href="#cite_note-BJS16-25">[25]</a>p. 26 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\varphi ({\mathsf {<}}\Omega ,0,\Omega +1))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif"><</mo> </mrow> </mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\varphi ({\mathsf {<}}\Omega ,0,\Omega +1))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4869213feb075ecada49330b4c5deb64cc9f93d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.104ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\varphi ({\mathsf {<}}\Omega ,0,\Omega +1))}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(U(ID))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">U</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(U(ID))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b0da7996de6097f32f468f1f3b5599797a12e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.134ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(U(ID))}}}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc7b885664ed612c07685fb9e230c9f2fa53252" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.309ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Omega _{\omega })}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{<\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{<\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/011f47d94d5ead7f3183bff84aa798b1b2b1434e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.858ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{<\omega }}" /><a href="#cite_note-RathjenFromArithmetic-4">[4]</a>p. 28 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3473e337c6f1f0bd6b050da016ef85c754c01102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.598ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA}}_{0}}" /><a href="#cite_note-RathjenFromArithmetic-4">[4]</a>p. 28, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c69c59f8dcd2de9bc6162361cd179610cb690d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.888ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}_{0}}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {MLW}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {MLW}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9d47ecc043b2f225c300799af4e0cc25e38997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.487ex; height:2.176ex;" alt="{\displaystyle {\mathsf {MLW}}}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {SUS}}+({\mathsf {S}}-{\mathsf {I}}_{\mathsf {N}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">S</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {SUS}}+({\mathsf {S}}-{\mathsf {I}}_{\mathsf {N}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7677926bea66bac1d38418c6eeda5dcb7b3ce9a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.009ex; height:2.843ex;" alt="{\displaystyle {\mathsf {SUS}}+({\mathsf {S}}-{\mathsf {I}}_{\mathsf {N}})}" /><a href="#cite_note-JagerStrahm02-34">[34]</a>p. 27 </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\omega }\omega ^{\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\omega }\omega ^{\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61bca83530e615b711f6a2e8bf193a30a5591666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.01ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Omega _{\omega }\omega ^{\omega })}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-IND}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-IND}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71b5a835f0148712a50519dd1d27e5ea77d6d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.112ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-IND}}}" /><a href="#cite_note-RathjenKrombholz2019-35">[35]</a> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\omega }\varepsilon _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\omega }\varepsilon _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f3a3640a926d9d554e94924af2d1d079eb2f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.447ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Omega _{\omega }\varepsilon _{0})}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {W-ID}}_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {W-ID}}_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d6cd6b9edee46d393f66b1c0a0c920c60a2c7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.614ex; height:2.509ex;" alt="{\displaystyle {\mathsf {W-ID}}_{\omega }}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3807538c4eeb801cebef3ed4105d30b0227ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.544ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA}}}" /><a href="#cite_note-36">[36]</a>p. 14 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {W-KPI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {W-KPI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef52d404fe0721f1485250c540e246548fda9dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.779ex; height:2.343ex;" alt="{\displaystyle {\mathsf {W-KPI}}}" /> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\omega }\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\omega }\Omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50277fb158c7564cd630ee0e067172b3de52e4fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.987ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Omega _{\omega }\Omega )}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA+BR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-CA+BR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a281140eabbc8e3eaeafacc0dc4bc8bf8b116ba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.533ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA+BR}}}" /><a href="#cite_note-37">[37]</a> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\omega }^{\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\omega }^{\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7825b3b98ec6a25a3bf3e0cb06102b64fd51f7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.309ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Omega _{\omega }^{\omega })}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c309174fcd09a1c87fcef72b6cb300f04e106c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.338ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-BI}}}" /><a href="#cite_note-RathjenKrombholz2019-35">[35]</a> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\omega }^{\omega ^{\omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\omega }^{\omega ^{\omega }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecee1ef1ec39ab39014f1fae02765587d324ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.304ex; height:3.176ex;" alt="{\displaystyle \psi _{0}(\Omega _{\omega }^{\omega ^{\omega }})}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-BI}}+\Pi _{3}^{1}{\mathsf {-IND}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-BI}}+\Pi _{3}^{1}{\mathsf {-IND}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32212082f7abda61d466f2cda083cf7ee5aba504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.755ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}+\Pi _{2}^{1}{\mathsf {-BI}}+\Pi _{3}^{1}{\mathsf {-IND}}}" /><a href="#cite_note-RathjenKrombholz2019-35">[35]</a> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9625b6e8398972bba3379799dc72e57bb3db714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_e">[5]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9046973c1d47ba06c6242d4cb863594933fca3d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.58ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{\omega }}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752c11d473cbfd10afa6edf5f2b14bf8d7b30978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.581ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Pi }}_{1}^{1}{\mathsf {-CA+BI}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ed2fad9bd3680aec68c9bcc7595de817ab34cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.745ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPI}}}" /> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\omega ^{\omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\omega ^{\omega }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad8b3e5fabba71c5d1eda42b90fcfe9cfdaa104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.304ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Omega _{\omega ^{\omega }})}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{<\omega ^{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{<\omega ^{\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242f91b582c7bda2f75cdb721efc946a68883153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.852ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{<\omega ^{\omega }}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f44b459e72fcdb69436bf485fbd518af4555ac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.785ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CR}}}" /><a href="#cite_note-RathjenFromArithmetic-4">[4]</a>p. 28 </td> <td> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {SUS}}+({\mathsf {N}}-{\mathsf {I}}_{\mathsf {N}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">S</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {SUS}}+({\mathsf {N}}-{\mathsf {I}}_{\mathsf {N}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e39ac848a656c36ef55736862443515b5431803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.362ex; height:2.843ex;" alt="{\displaystyle {\mathsf {SUS}}+({\mathsf {N}}-{\mathsf {I}}_{\mathsf {N}})}" /><a href="#cite_note-JagerStrahm02-34">[34]</a>p. 27 </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\varepsilon _{0}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\varepsilon _{0}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e0293864d1efff0e3acf52529b452d395580d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.885ex; height:3.009ex;" alt="{\displaystyle \psi _{0}(\Omega _{\varepsilon _{0}})}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{<\varepsilon _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{<\varepsilon _{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3302d79cb3933094d010cca46574b86116c740f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.433ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ID}}_{<\varepsilon _{0}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87c51209075cc2006bb099b0897cfffed3eec09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.834ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}}" /><a href="#cite_note-RathjenFromArithmetic-4">[4]</a>p. 28, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Sigma }}_{2}^{1}{\mathsf {-AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Sigma }}_{2}^{1}{\mathsf {-AC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1958b8cf6e815abdf191548f7733620dfef570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.576ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Sigma }}_{2}^{1}{\mathsf {-AC}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {W-KPi}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {W-KPi}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1537b1fe3440ffcfdb16ec050284c992821998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.689ex; height:2.343ex;" alt="{\displaystyle {\mathsf {W-KPi}}}" /> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {SUS}}+(\mathrm {L} -{\mathsf {I}}_{\mathsf {N}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">S</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {SUS}}+(\mathrm {L} -{\mathsf {I}}_{\mathsf {N}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597291f1c3e4e3adb4fe48e5868b70cffa3fb995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.17ex; height:2.843ex;" alt="{\displaystyle {\mathsf {SUS}}+(\mathrm {L} -{\mathsf {I}}_{\mathsf {N}})}" /><a href="#cite_note-JagerStrahm02-34">[34]</a>p. 27 </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Omega _{\Omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Omega _{\Omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c3b78ce9c40c6c077fc396757e890ac83928eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.474ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Omega _{\Omega })}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(ID)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(ID)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3674e73fe6b4f04367b7a6830558209a673b004e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.726ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(ID)}}}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_g">[6]</a> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\Omega _{1}}(\varepsilon _{\Omega _{\Omega }+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\Omega _{1}}(\varepsilon _{\Omega _{\Omega }+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b9a4ab601c99f2a2f2418fc7c42b4160d4935d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.303ex; height:3.009ex;" alt="{\displaystyle \psi _{\Omega _{1}}(\varepsilon _{\Omega _{\Omega }+1})}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{\prec ^{*}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mo>≺</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{\prec ^{*}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b2624c2212a8374685d77a448beb5b7a0cc632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.667ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{\prec ^{*}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {BID}}^{2*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {BID}}^{2*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e44f205b459893e591217f23776d0ccb62414ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.676ex;" alt="{\displaystyle {\mathsf {BID}}^{2*}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}^{2*}+{\mathsf {BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>∗</mo> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}^{2*}+{\mathsf {BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2200191de19f996f7cf17b1b017f022c9efbb57d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.239ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ID}}^{2*}+{\mathsf {BI}}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPl}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPl}}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ba8f3f250f82ab03183ad0928b88a5e1571d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.709ex; height:2.343ex;" alt="{\displaystyle {\mathsf {KPl}}^{*}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPl}}_{\Omega }^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPl}}_{\Omega }^{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e136e17e2eb14ed518f5b42f7428a4d41f3629ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.073ex; height:2.843ex;" alt="{\displaystyle {\mathsf {KPl}}_{\Omega }^{r}}" /> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Phi _{1}(0))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Phi _{1}(0))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11c446c7d3f682132f28a3e170c7541172757e66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.081ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Phi _{1}(0))}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5289c31efa8a7e62b225a027a13c1aa279fed29a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.744ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}_{0}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}_{0}+\Delta _{2}^{1}{\mathsf {-CA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}_{0}+\Delta _{2}^{1}{\mathsf {-CA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81339c89b25e92809fc47c205e5a50f500b8e917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.473ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}_{0}+\Delta _{2}^{1}{\mathsf {-CA}}_{0}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA+BI(impl-}}\Sigma _{2}^{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">l</mi> <mo mathvariant="sans-serif">−</mo> </mrow> </mrow> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}{\mathsf {-CA+BI(impl-}}\Sigma _{2}^{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a56edc60df78fa0170e92a7de93c5abc40fb03f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.381ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA+BI(impl-}}\Sigma _{2}^{1})}" />,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA+BR(impl-}}\Sigma _{2}^{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">R</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">l</mi> <mo mathvariant="sans-serif">−</mo> </mrow> </mrow> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}{\mathsf {-CA+BR(impl-}}\Sigma _{2}^{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc970dd892cce24015505f35640b041f5831db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.236ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA+BR(impl-}}\Sigma _{2}^{1})}" />,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-ID} _{0}^{pos}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>o</mi> <mi>s</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-ID} _{0}^{pos}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c93c2b7cf26142b9d7c0a3ae5c55e3eda1da36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.736ex; height:3.176ex;" alt="{\displaystyle \mathbf {AUT-ID} _{0}^{pos}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-ID} _{0}^{mon}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>o</mi> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-ID} _{0}^{mon}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f54b5c86ce9892114a6e8a6faea2bbd37fda018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.567ex; height:2.843ex;" alt="{\displaystyle \mathbf {AUT-ID} _{0}^{mon}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}^{w}+{\mathsf {FOUNDR}}({\mathsf {impl-}})\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">O</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">l</mi> <mo mathvariant="sans-serif">−</mo> </mrow> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}^{w}+{\mathsf {FOUNDR}}({\mathsf {impl-}})\Sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597ec76c6e049ca52b969c9ba0a6c75403d4fe59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.721ex; height:2.843ex;" alt="{\displaystyle {\mathsf {KPi}}^{w}+{\mathsf {FOUNDR}}({\mathsf {impl-}})\Sigma )}" />,<a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}^{w}+{\mathsf {FOUND}}({\mathsf {impl-}})\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">O</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">l</mi> <mo mathvariant="sans-serif">−</mo> </mrow> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}^{w}+{\mathsf {FOUND}}({\mathsf {impl-}})\Sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3251698bf0f47ff23bf20fe4db98745a20e78743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.22ex; height:2.843ex;" alt="{\displaystyle {\mathsf {KPi}}^{w}+{\mathsf {FOUND}}({\mathsf {impl-}})\Sigma )}" />,<a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-KPl} ^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-KPl} ^{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547c9d463f2a281ee444d7dce6499d95938365b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.682ex; height:2.343ex;" alt="{\displaystyle \mathbf {AUT-KPl} ^{r}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-KPl} ^{r}+\mathbf {KPi} ^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-KPl} ^{r}+\mathbf {KPi} ^{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72fa1bf034b63f1731d61aaef122eb577e220d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.158ex; height:2.509ex;" alt="{\displaystyle \mathbf {AUT-KPl} ^{r}+\mathbf {KPi} ^{r}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Phi _{1}(0)\varepsilon _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Phi _{1}(0)\varepsilon _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80023db5797d379a7a3c4444fa95b6abf3deef99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.218ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Phi _{1}(0)\varepsilon _{0})}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce04a315fad26cd7ced9e5b8b29e7369bda671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.69ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-ID} ^{pos}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>o</mi> <mi>s</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-ID} ^{pos}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c573cc1c122a3c94cac0f9778fe92ae050e2d86e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.736ex; height:2.343ex;" alt="{\displaystyle \mathbf {AUT-ID} ^{pos}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-ID} ^{mon}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-ID} ^{mon}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abe321f6b6e27d18b8c185f04fda67371578b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.567ex; height:2.343ex;" alt="{\displaystyle \mathbf {AUT-ID} ^{mon}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-KPl} ^{w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-KPl} ^{w}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3f25748b928215e6fca92bdd7f0b2cf5ee72f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.117ex; height:2.343ex;" alt="{\displaystyle \mathbf {AUT-KPl} ^{w}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\varepsilon _{\Phi _{1}(0)+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\varepsilon _{\Phi _{1}(0)+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f35e02d198f795f4b9a0b7e8d9a1eb3893c864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.912ex; height:3.176ex;" alt="{\displaystyle \psi _{0}(\varepsilon _{\Phi _{1}(0)+1})}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+({\mathsf {BI}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+({\mathsf {BI}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2039575d9511c50cac1d71417d407fdce2d5affc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.537ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+({\mathsf {BI}})}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-ID} _{2}^{pos}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>o</mi> <mi>s</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-ID} _{2}^{pos}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddccb2851c2051eb1f2f0884dc775701c617ef2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.736ex; height:3.176ex;" alt="{\displaystyle \mathbf {AUT-ID} _{2}^{pos}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-ID} _{2}^{mon}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">I</mi> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>o</mi> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-ID} _{2}^{mon}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2360977e97a744a424fa1d89752d8550c0d66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.567ex; height:2.843ex;" alt="{\displaystyle \mathbf {AUT-ID} _{2}^{mon}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-KPl} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-KPl} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efeecc6ece39345ba1d09e952a0ef3a14b265f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.708ex; height:2.176ex;" alt="{\displaystyle \mathbf {AUT-KPl} }" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Phi _{1}(\varepsilon _{0}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Phi _{1}(\varepsilon _{0}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be15bc88c24b48f5a4cebdb8a106b1886ec073ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.056ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Phi _{1}(\varepsilon _{0}))}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+\Delta _{2}^{1}{\mathsf {-CA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+\Delta _{2}^{1}{\mathsf {-CA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310e335239dee7b2da4cdd9a3042e90a05e566eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.364ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+\Delta _{2}^{1}{\mathsf {-CA}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+\Sigma _{2}^{1}{\mathsf {-AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+\Sigma _{2}^{1}{\mathsf {-AC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48566fe6b5a1d96876651c2e82baad74c496222d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.106ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}{\mathsf {-TR}}+\Sigma _{2}^{1}{\mathsf {-AC}}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AUT-KPl} ^{w}+\mathbf {KPi} ^{w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">T</mi> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AUT-KPl} ^{w}+\mathbf {KPi} ^{w}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad153aa815e97eff9c691c047f6848de380746cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.029ex; height:2.509ex;" alt="{\displaystyle \mathbf {AUT-KPl} ^{w}+\mathbf {KPi} ^{w}}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Phi _{\omega }(0))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Phi _{\omega }(0))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149ffdcac7027e90df63147fe5915181019c47c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.281ex; height:2.843ex;" alt="{\displaystyle \psi _{0}(\Phi _{\omega }(0))}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}{\mathsf {-TR}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}{\mathsf {-TR}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c55ce97e03efc472f9512d244872d6118a36eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.937ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}{\mathsf {-TR}}_{0}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{2}^{1}{\mathsf {-TRDC}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{2}^{1}{\mathsf {-TRDC}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0dfd5787b5a476dab190820f194c6a3aa8cd4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.842ex; height:3.176ex;" alt="{\displaystyle \Sigma _{2}^{1}{\mathsf {-TRDC}}_{0}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA}}_{0}+(\Sigma _{2}^{1}{\mathsf {-BI}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}{\mathsf {-CA}}_{0}+(\Sigma _{2}^{1}{\mathsf {-BI}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc6a2362e9253023fdb7109b956b190d90b53e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.275ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA}}_{0}+(\Sigma _{2}^{1}{\mathsf {-BI}})}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {KPi} ^{r}+(\Sigma {\mathsf {-FOUND}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">O</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {KPi} ^{r}+(\Sigma {\mathsf {-FOUND}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bd4c01886d7cc52127e932e02b5ad95b13f263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.728ex; height:2.843ex;" alt="{\displaystyle \mathbf {KPi} ^{r}+(\Sigma {\mathsf {-FOUND}})}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {KPi} ^{r}+(\Sigma {\mathsf {-REC}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {KPi} ^{r}+(\Sigma {\mathsf {-REC}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2408d42cd66f47b3ab68bcf5091936efaad3a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.147ex; height:2.843ex;" alt="{\displaystyle \mathbf {KPi} ^{r}+(\Sigma {\mathsf {-REC}})}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(\Phi _{\varepsilon _{0}}(0))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(\Phi _{\varepsilon _{0}}(0))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8d5960e15aa158e5442636eb92bac967e3f66d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.856ex; height:3.009ex;" alt="{\displaystyle \psi _{0}(\Phi _{\varepsilon _{0}}(0))}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}{\mathsf {-TR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}{\mathsf {-TR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb6a04a4cac0fdd5c28995530f03b39d24d4b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.883ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}{\mathsf {-TR}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{2}^{1}{\mathsf {-TRDC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{2}^{1}{\mathsf {-TRDC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a0ca7f7c331bda0989bac30f2e87da2b7b1ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.788ex; height:3.176ex;" alt="{\displaystyle \Sigma _{2}^{1}{\mathsf {-TRDC}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA}}+(\Sigma _{2}^{1}{\mathsf {-BI}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}{\mathsf {-CA}}+(\Sigma _{2}^{1}{\mathsf {-BI}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35437b2fafc865c77e1ac22fc552516cf4f5bf75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.221ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}{\mathsf {-CA}}+(\Sigma _{2}^{1}{\mathsf {-BI}})}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {KPi} ^{w}+(\Sigma {\mathsf {-FOUND}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">O</mi> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {KPi} ^{w}+(\Sigma {\mathsf {-FOUND}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85113c6ecb1723b96b01ddee3766e0513acdfe6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.164ex; height:2.843ex;" alt="{\displaystyle \mathbf {KPi} ^{w}+(\Sigma {\mathsf {-FOUND}})}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {KPi} ^{w}+(\Sigma {\mathsf {-REC}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {KPi} ^{w}+(\Sigma {\mathsf {-REC}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb9317b02cb3a5850345388d53b168be4ca3c60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.582ex; height:2.843ex;" alt="{\displaystyle \mathbf {KPi} ^{w}+(\Sigma {\mathsf {-REC}})}" /><a href="#cite_note-RathjenInvestigations-38">[38]</a>: 72  </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\varepsilon _{I+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\varepsilon _{I+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514cef023ca0d237f8161bcaee5999f26efc091a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.567ex; height:2.843ex;" alt="{\displaystyle \psi (\varepsilon _{I+1})}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_h">[7]</a> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9520ed39ec648af1ae0640d051ea497774381c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.871ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI}}}" /><a href="#cite_note-RathjenFromArithmetic-4">[4]</a>p. 28, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Sigma }}_{2}^{1}{\mathsf {-AC+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Sigma }}_{2}^{1}{\mathsf {-AC+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953a2654b2cb9c1874cbddec9bd7d9bdc1084d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.614ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Sigma }}_{2}^{1}{\mathsf {-AC+BI}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830e97ea8ec435efbeea4e4005b3127c08799cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPi}}}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF+REA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF+REA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca379dbbbec7fb23f6550147a48e72262c3aca38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.508ex; height:2.343ex;" alt="{\displaystyle {\mathsf {CZF+REA}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ab81b830d372ce7c5d1be9545f4710a34a0279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle {\mathsf {T}}_{0}}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\Omega _{I+\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>+</mo> <mi>ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\Omega _{I+\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1505db94b19b244a5175cde8c44a772146dbb440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.362ex; height:2.843ex;" alt="{\displaystyle \psi (\Omega _{I+\omega })}" /> </td> <td> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{1}{\mathsf {W}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed399b8e22e64a92f8ab8b3d8e7be24ae2d20ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.541ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {W}}}" /><a href="#cite_note-39">[39]</a>: 38  </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\Omega _{L})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\Omega _{L})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d8ac6f5deee578e0c3ce114c9f82ce1842e2c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.352ex; height:2.843ex;" alt="{\displaystyle \psi (\Omega _{L})}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_i">[8]</a> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPh}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPh}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1146ae4affa7cdac8e348380e4b7ab10a6e6b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.3ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPh}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{<\omega }{\mathsf {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{<\omega }{\mathsf {W}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071d3bb9b4492129e3cdfedae8ff1ccef298b42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.02ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{<\omega }{\mathsf {W}}}" /> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\Omega _{L^{*}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\Omega _{L^{*}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c68849e5b3f88081c9c598a76cc2ea5beed34d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.184ex; height:2.843ex;" alt="{\displaystyle \psi (\Omega _{L^{*}})}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_j">[9]</a> </td> <td> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(MLW)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(MLW)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233474cc9bd8b8ee7a3bcda02553dad198993259" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.888ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(MLW)}}}" /> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\Omega }(\chi _{\varepsilon _{M+1}}(0))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\Omega }(\chi _{\varepsilon _{M+1}}(0))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad1edfd2fd820153a355337e6a04c8c63095ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.438ex; height:3.009ex;" alt="{\displaystyle \psi _{\Omega }(\chi _{\varepsilon _{M+1}}(0))}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_k">[10]</a> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI+(M)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> <mo mathvariant="sans-serif">+</mo> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">M</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI+(M)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0099b833de697e060a36ecca392431e87b519d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.555ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI+(M)}}}" /><a href="#cite_note-40">[40]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPM}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbebfc6d39aa000417f7935132cfe0724795a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.132ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPM}}}" /> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZFM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZFM}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e084eeb1df10cdae337b468508c7c8d3846e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.262ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CZFM}}}" /> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\Omega _{M+\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>+</mo> <mi>ω</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\Omega _{M+\omega })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/080cc5401383eeec9cda041620e804edb68844d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.261ex; height:2.843ex;" alt="{\displaystyle \psi (\Omega _{M+\omega })}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_o">[11]</a> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPM}}^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPM}}^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c6f2cff8c5e90582fda564ac6c248b0806be27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.643ex; height:2.509ex;" alt="{\displaystyle {\mathsf {KPM}}^{+}}" /><a href="#cite_note-Setzer96-41">[41]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {TTM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {TTM}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d18184b98126bd325a8d7485a8c421f10260dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.199ex; height:2.176ex;" alt="{\displaystyle {\mathsf {TTM}}}" /><a href="#cite_note-Setzer96-41">[41]</a> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\Omega }^{0}(\varepsilon _{K+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{\Omega }^{0}(\varepsilon _{K+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfcc4865cad4e46a97f85297eb716e6414571fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.913ex; height:3.176ex;" alt="{\displaystyle \Psi _{\Omega }^{0}(\varepsilon _{K+1})}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_l">[12]</a> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP+\Pi }}_{3}-{\mathsf {Ref}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">Π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP+\Pi }}_{3}-{\mathsf {Ref}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0446f656441579b9e9360bdf904695d7d2f6fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.82ex; height:2.509ex;" alt="{\displaystyle {\mathsf {KP+\Pi }}_{3}-{\mathsf {Ref}}}" /><a href="#cite_note-42">[42]</a> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{(\omega ^{+};P_{0},\epsilon ,\epsilon ,0)}^{\varepsilon _{\Xi +1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>;</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>ϵ</mi> <mo>,</mo> <mi>ϵ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ξ</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{(\omega ^{+};P_{0},\epsilon ,\epsilon ,0)}^{\varepsilon _{\Xi +1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e07d3d37705819b3711be81ab0ca5630aefa80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:11.418ex; height:3.676ex;" alt="{\displaystyle \Psi _{(\omega ^{+};P_{0},\epsilon ,\epsilon ,0)}^{\varepsilon _{\Xi +1}}}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_m">[13]</a> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP+\Pi }}_{\omega }-{\mathsf {Ref}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">Π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP+\Pi }}_{\omega }-{\mathsf {Ref}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d912f87d93b6edcb9e590a3fb9eccdc64c87c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.02ex; height:2.509ex;" alt="{\displaystyle {\mathsf {KP+\Pi }}_{\omega }-{\mathsf {Ref}}}" /><a href="#cite_note-Stegert10-43">[43]</a> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{(\omega ^{+};P_{0},\epsilon ,\epsilon ,0)}^{\varepsilon _{\Upsilon +1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>;</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>ϵ</mi> <mo>,</mo> <mi>ϵ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Υ</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{(\omega ^{+};P_{0},\epsilon ,\epsilon ,0)}^{\varepsilon _{\Upsilon +1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e49bc546f58876f4703637ba130bc19e05b266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:11.418ex; height:3.843ex;" alt="{\displaystyle \Psi _{(\omega ^{+};P_{0},\epsilon ,\epsilon ,0)}^{\varepsilon _{\Upsilon +1}}}" /><a class="external autonumber" href="https://en.wikipedia.org/wiki/Ordinal_analysis#endnote_n">[14]</a> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Stability}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">b</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">y</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Stability}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ad48acf3b8e5c9d3596e36eb7b588569eae57e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.033ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Stability}}}" /><a href="#cite_note-Stegert10-43">[43]</a> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {S} ^{+}+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>K</mi> </mrow> </msubsup> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {S} ^{+}+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf712028289173945cae751d81bcc11418ba1b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.474ex; height:3.676ex;" alt="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {S} ^{+}+1})}" /><a href="#cite_note-Arai23A-44">[44]</a> </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}\omega +\Pi _{1}^{1}-{\mathsf {Ref}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mi>ω</mi> <mo>+</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}\omega +\Pi _{1}^{1}-{\mathsf {Ref}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f0a2c0e599bc83480232b83a8101a9122c79d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.363ex; height:3.176ex;" alt="{\displaystyle {\mathsf {KP}}\omega +\Pi _{1}^{1}-{\mathsf {Ref}}}" />,<a href="#cite_note-Arai23A-44">[44]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}\omega +(M\prec _{\Sigma _{1}}V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mi>ω</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>M</mi> <msub> <mo>≺</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}\omega +(M\prec _{\Sigma _{1}}V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0592e7218df0a9127e701c3d1e641507141f000e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.772ex; height:3.009ex;" alt="{\displaystyle {\mathsf {KP}}\omega +(M\prec _{\Sigma _{1}}V)}" /><a href="#cite_note-Arai23B-45">[45]</a> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} +1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>K</mi> </mrow> </msubsup> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} +1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20bad1b8ada25a00ca496d1875b779ef17614d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.997ex; height:3.676ex;" alt="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} +1})}" /><a href="#cite_note-Arai23A-44">[44]</a> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{3}^{1}{\mathsf {-DC+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{3}^{1}{\mathsf {-DC+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/123df3aa0109ced0e633242540a46500ad50dac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.741ex; height:3.176ex;" alt="{\displaystyle \Sigma _{3}^{1}{\mathsf {-DC+BI}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{3}^{1}{\mathsf {-AC+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{3}^{1}{\mathsf {-AC+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6edf1d52c4a3a652ceef761e43eea2790895df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.614ex; height:3.176ex;" alt="{\displaystyle \Sigma _{3}^{1}{\mathsf {-AC+BI}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}\omega +\Pi _{1}-{\mathsf {Collection}}+(V=L)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mi>ω</mi> <mo>+</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>=</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}\omega +\Pi _{1}-{\mathsf {Collection}}+(V=L)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05dabff1bdbbb78c866deec1e63dae49afef160f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.726ex; height:2.843ex;" alt="{\displaystyle {\mathsf {KP}}\omega +\Pi _{1}-{\mathsf {Collection}}+(V=L)}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{N}+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>K</mi> </mrow> </msubsup> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{N}+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b0fd8223002ab0887d74c555b0bc3c203f63f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.346ex; height:3.676ex;" alt="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{N}+1})}" /><a href="#cite_note-Arai24-46">[46]</a> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{N+2}^{1}{\mathsf {-DC+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">C</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{N+2}^{1}{\mathsf {-DC+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedfba518598fee054d5418b36dcd0a74d65786a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.479ex; height:3.343ex;" alt="{\displaystyle \Sigma _{N+2}^{1}{\mathsf {-DC+BI}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{N+2}^{1}{\mathsf {-AC+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{N+2}^{1}{\mathsf {-AC+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40260a626f4a5995fc8b8603cc6fb6bb446e4d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.351ex; height:3.343ex;" alt="{\displaystyle \Sigma _{N+2}^{1}{\mathsf {-AC+BI}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mi>ω</mi> <mo>+</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>=</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38bea4ba8a3d6a33edf986a46eeb7590058d55c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.363ex; height:2.843ex;" alt="{\displaystyle {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)}" /> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td>? </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}+\bigcup \limits _{N<\omega }{\mathsf {TI}}[\Pi _{0}^{1-},\psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{N}+1})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mo>+</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo><</mo> <mi>ω</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> </mrow> </msubsup> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>K</mi> </mrow> </msubsup> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}+\bigcup \limits _{N<\omega }{\mathsf {TI}}[\Pi _{0}^{1-},\psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{N}+1})]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5bcee0ec9a0692e1322c8aa2c967e770be2952f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.003ex; height:5.676ex;" alt="{\displaystyle {\mathsf {PA}}+\bigcup \limits _{N<\omega }{\mathsf {TI}}[\Pi _{0}^{1-},\psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{N}+1})]}" /><a href="#cite_note-Arai24-46">[46]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3b518a93ad59911a077e8f316a4ea4aebc870c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.688ex; height:2.509ex;" alt="{\displaystyle \mathbf {Z} _{2}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{\infty }^{1}-{\mathsf {CA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{\infty }^{1}-{\mathsf {CA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8431757a5e524d0f266e8906360607958c65a723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.495ex; height:2.843ex;" alt="{\displaystyle \Pi _{\infty }^{1}-{\mathsf {CA}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}+\Pi _{\omega }^{\text{set}}-{\mathsf {Separation}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>set</mtext> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">p</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}+\Pi _{\omega }^{\text{set}}-{\mathsf {Separation}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6ec7f23e53bb252987ad34ee02f0c368c85561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.091ex; height:2.676ex;" alt="{\displaystyle {\mathsf {KP}}+\Pi _{\omega }^{\text{set}}-{\mathsf {Separation}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38b84807ebde16d0af8fc1ffa9500957b272bd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.518ex; height:2.176ex;" alt="{\displaystyle \lambda 2}" /><a href="#cite_note-47">[47]</a> </td> <td> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Key">Key</h3>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=16" title="Edit section: Key">edit</a>]</div> This is a list of symbols used in this table: <ul><li>ψ represents various <a href="/wiki/Ordinal_collapsing_function" title="Ordinal collapsing function">ordinal collapsing functions</a> as defined in their respective citations.</li> <li>Ψ represents either Rathjen's or Stegert's Psi.</li> <li>φ represents Veblen's function.</li> <li>ω represents the first transfinite ordinal.</li> <li>εα represents the <a href="/wiki/Epsilon_numbers_(mathematics)" class="mw-redirect" title="Epsilon numbers (mathematics)">epsilon numbers</a>.</li> <li>Γα represents the gamma numbers (Γ0 is the <a href="/wiki/Feferman%E2%80%93Sch%C3%BCtte_ordinal" title="Feferman–Schütte ordinal">Feferman–Schütte ordinal</a>)</li> <li>Ωα represent the uncountable ordinals (Ω1, abbreviated Ω, is <a href="/wiki/First_uncountable_ordinal" title="First uncountable ordinal">ω1</a>). Countability is considered necessary for an ordinal to be regarded as proof theoretic.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" /> is an ordinal term denoting a stable ordinal, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca04aede733c0129f157a88c04c76c7ef2afe28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.803ex; height:2.509ex;" alt="{\displaystyle \mathbb {S} ^{+}}" /> the least admissible ordinal above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" />.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {I} _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {I} _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28136f96bded213af8d3bff043d924c97b47c2a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.596ex; height:2.509ex;" alt="{\displaystyle \mathbb {I} _{N}}" /> is an ordinal term denoting an ordinal such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\mathbb {I} _{N}}\models {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo>⊨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mi>ω</mi> <mo>+</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>=</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\mathbb {I} _{N}}\models {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd566a658e8316697b876ac7b3302e6c20ac1a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.472ex; height:3.009ex;" alt="{\displaystyle L_{\mathbb {I} _{N}}\models {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)}" />; N is a variable that defines a series of ordinal analyses of the results of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{N}-{\mathsf {Collection}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{N}-{\mathsf {Collection}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00a47ef9517d7a08d83b7afcfa7e609749ee702c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.86ex; height:2.509ex;" alt="{\displaystyle \Pi _{N}-{\mathsf {Collection}}}" /> forall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq N<\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>N</mi> <mo><</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq N<\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b369c8a06720a1dbf158bf56aa130d798c3f5b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.869ex; height:2.343ex;" alt="{\displaystyle 1\leq N<\omega }" />. when N=1, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{1}+1})=\psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} +1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>K</mi> </mrow> </msubsup> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>K</mi> </mrow> </msubsup> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{1}+1})=\psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} +1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ad46a3a0599036282cc5e102d3b0ddf75e7491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.925ex; height:3.676ex;" alt="{\displaystyle \psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} _{1}+1})=\psi _{\omega _{1}^{CK}}(\varepsilon _{\mathbb {I} +1})}" /></li></ul> This is a list of the abbreviations used in this table: <ul><li>First-order arithmetic <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b954e6612a5bc83e108fd4eda3c54901a2786f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.711ex; height:2.343ex;" alt="{\displaystyle {\mathsf {Q}}}" /> is <a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798737a1975f160943f3b8bef6bb1d5ff72a4779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.546ex; height:2.509ex;" alt="{\displaystyle {\mathsf {PA}}^{-}}" /> is the first-order theory of the nonnegative part of a discretely ordered ring.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RFA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RFA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62fa1b178e47c519061f20c0ca3c76ba23b17f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.375ex; height:2.176ex;" alt="{\displaystyle {\mathsf {RFA}}}" /> is <a href="/wiki/Jensen_hierarchy" title="Jensen hierarchy">rudimentary function</a> arithmetic.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Delta }}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Delta }}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b83607f7f952ec79ef9fd3e3fc5c939c1c406f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.637ex; height:2.509ex;" alt="{\displaystyle {\mathsf {I\Delta }}_{0}}" /> is arithmetic with induction restricted to Δ0-predicates without any axiom asserting that exponentiation is total.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EFA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EFA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f822e132414d381dc8bae0ec41097739a72f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.261ex; height:2.176ex;" alt="{\displaystyle {\mathsf {EFA}}}" /> is <a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function arithmetic</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4913b90578aec06daa7907274083a49f0d4063b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.094ex; height:3.176ex;" alt="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}" /> is arithmetic with induction restricted to Δ0-predicates augmented by an axiom asserting that exponentiation is total.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EFA}}^{\mathsf {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EFA}}^{\mathsf {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05790454736255100deb5ccd8b2d197f7d053a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.343ex; height:2.343ex;" alt="{\displaystyle {\mathsf {EFA}}^{\mathsf {n}}}" /> is elementary function arithmetic augmented by an axiom ensuring that each element of the n-th level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f57d70ee7aeb520dba041fbb5edd2bbce74378cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.555ex; height:2.343ex;" alt="{\displaystyle {\mathcal {E}}^{n}}" /> of the <a href="/wiki/Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a> is total.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mo mathvariant="sans-serif">+</mo> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2ea8f071f17c5881b2629d44249ff8437db781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.943ex; height:3.176ex;" alt="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4913b90578aec06daa7907274083a49f0d4063b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.094ex; height:3.176ex;" alt="{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}}" /> augmented by an axiom ensuring that each element of the n-th level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f57d70ee7aeb520dba041fbb5edd2bbce74378cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.555ex; height:2.343ex;" alt="{\displaystyle {\mathcal {E}}^{n}}" /> of the <a href="/wiki/Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a> is total.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PRA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PRA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc753f99fc799cd0ee9fb4c163fd66a551e4d653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.537ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PRA}}}" /> is <a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive arithmetic</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {I\Sigma }}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Σ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {I\Sigma }}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3e869994d0704d1d11236dc6abb77d3f2952c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:2.509ex;" alt="{\displaystyle {\mathsf {I\Sigma }}_{1}}" /> is arithmetic with induction restricted to Σ1-predicates.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {PA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {PA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6812979524cbdd07e79311aa1263ab6db58758f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle {\mathsf {PA}}}" /> is <a href="/wiki/Peano_axioms" title="Peano axioms">Peano arithmetic</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{\nu }\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{\nu }\#}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e712f7e4439ebcacfcf332f510e8daba6d456821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.364ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{\nu }\#}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e937cf9a67178dabc08089a62218b47f7ed5f624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:3.438ex; height:3.343ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}" /> but with induction only for positive formulas.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e937cf9a67178dabc08089a62218b47f7ed5f624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:3.438ex; height:3.343ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}" /> extends PA by ν iterated fixed points of monotone operators.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {U(PA)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">U</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {U(PA)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9ed8c5ddffdf4f7c8573b1c39193b8d5481159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:2.843ex;" alt="{\displaystyle {\mathsf {U(PA)}}}" /> is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on the natural numbers.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut({\widehat {ID}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo mathvariant="sans-serif">^</mo> </mover> </mrow> </mrow> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut({\widehat {ID}})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ab492ed2e6ea464187f0b6f460d80430f26166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.726ex; height:3.509ex;" alt="{\displaystyle {\mathsf {Aut({\widehat {ID}})}}}" /> is autonomously iterated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e937cf9a67178dabc08089a62218b47f7ed5f624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:3.438ex; height:3.343ex;" alt="{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }}" /> (in other words, once an ordinal is defined, it can be used to index a new series of definitions.)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180426fbf935cba02bd20d299b9d25218abfdc20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{\nu }}" /> extends PA by ν iterated least fixed points of monotone operators.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">U</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8685b02ac8066d390c53cc592df4175801f640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.837ex; height:3.009ex;" alt="{\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}}" /> is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on ν-times iterated generalized inductive definitions.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(U(ID))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">U</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(U(ID))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b0da7996de6097f32f468f1f3b5599797a12e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.134ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(U(ID))}}}" /> is autonomously iterated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">U</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8685b02ac8066d390c53cc592df4175801f640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.837ex; height:3.009ex;" alt="{\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}}" />.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {W-ID}}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {W-ID}}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7181314f6c0a3e3be55e53302d17af3a7bceda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.463ex; height:2.509ex;" alt="{\displaystyle {\mathsf {W-ID}}_{\nu }}" /> is a weakened version of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ID}}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ID}}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180426fbf935cba02bd20d299b9d25218abfdc20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ID}}_{\nu }}" /> based on W-types.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {TI}}[\Pi _{0}^{1-},\alpha ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> </mrow> </msubsup> <mo>,</mo> <mi>α</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {TI}}[\Pi _{0}^{1-},\alpha ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab99a1ed8c6503eb1ce734783eed99e63461547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.121ex; height:3.343ex;" alt="{\displaystyle {\mathsf {TI}}[\Pi _{0}^{1-},\alpha ]}" /> is a transfinite induction of length α no more than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{0}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae152f64cb76d9503f9a85e10d72902114b12c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{0}^{1}}" />-formulas. It happens to be the representation of the ordinal notation when used in first-order arithmetic.</li></ul></li></ul> <ul><li>Second-order arithmetic</li></ul> In general, a subscript 0 means that the induction scheme is restricted to a single set induction axiom. <ul><li><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}_{0}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}_{0}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/387dd1e66e5c4f3d18d3cbc2c904866b74837c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.843ex;" alt="{\displaystyle {\mathsf {RCA}}_{0}^{*}}" /> is a second order form of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EFA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EFA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f822e132414d381dc8bae0ec41097739a72f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.261ex; height:2.176ex;" alt="{\displaystyle {\mathsf {EFA}}}" /> sometimes used in <a href="/wiki/Reverse_mathematics" title="Reverse mathematics">reverse mathematics</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {WKL}}_{0}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {WKL}}_{0}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7da1ced71d326bc9f74da8ee40b7aab8f7652c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.121ex; height:2.843ex;" alt="{\displaystyle {\mathsf {WKL}}_{0}^{*}}" /> is a second order form of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EFA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EFA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f822e132414d381dc8bae0ec41097739a72f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.261ex; height:2.176ex;" alt="{\displaystyle {\mathsf {EFA}}}" /> sometimes used in reverse mathematics.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {RCA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {RCA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25465d37b9757defbfdec0ceae66f5281b79ae2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.591ex; height:2.509ex;" alt="{\displaystyle {\mathsf {RCA}}_{0}}" /> is <a href="/wiki/Second-order_arithmetic#Recursive_comprehension" title="Second-order arithmetic">recursive comprehension</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {WKL}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {WKL}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f2e0165d0148181304ad8f47791d2a494d7417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.121ex; height:2.509ex;" alt="{\displaystyle {\mathsf {WKL}}_{0}}" /> is <a href="/wiki/Reverse_mathematics#Weak_Kőnig's_lemma_WKL0" title="Reverse mathematics">weak Kőnig's lemma</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c859a47f5d58a4af830b235a0852a326b9febca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.64ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ACA}}_{0}}" /> is <a href="/wiki/Second-order_arithmetic#Arithmetical_comprehension" title="Second-order arithmetic">arithmetical comprehension</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a91137e338ef377365b38f8c19d94d5dc11b618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.586ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ACA}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ACA}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ACA}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c859a47f5d58a4af830b235a0852a326b9febca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.64ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ACA}}_{0}}" /> plus the full second-order induction scheme.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee518e00cc8bc05f29a182eaf59765c1ef7624b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.689ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ATR}}_{0}}" /> is <a href="/wiki/Reverse_mathematics#Arithmetical_transfinite_recursion_ATR0" title="Reverse mathematics">arithmetical transfinite recursion</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/833b7bcdd3762b701897bd2d3f2a0b718416124d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.635ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ATR}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ATR}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ATR}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee518e00cc8bc05f29a182eaf59765c1ef7624b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.689ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ATR}}_{0}}" /> plus the full second-order induction scheme.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI+(M)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> <mo mathvariant="sans-serif">+</mo> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">M</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI+(M)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0099b833de697e060a36ecca392431e87b519d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.555ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI+(M)}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9520ed39ec648af1ae0640d051ea497774381c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.871ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA+BI}}}" /> plus the assertion "every true <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Pi }}_{3}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Pi }}_{3}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5dd7a3e32968f8134eab636af50dfb1c0b6ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.7ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Pi }}_{3}^{1}}" />-sentence with parameters holds in a (countable coded) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" />-model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Δ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87c51209075cc2006bb099b0897cfffed3eec09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.834ex; height:3.176ex;" alt="{\displaystyle {\mathsf {\Delta }}_{2}^{1}{\mathsf {-CA}}}" />".</li></ul></li></ul> <ul><li>Kripke-Platek set theory <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94295759f4f38c35275d7f87f61ec91690ef5028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KP}}}" /> is <a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke-Platek set theory</a> with the axiom of infinity.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi>ω</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da18b92636475f92b90c49e811e91b0ee17d64c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.544ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KP\omega }}}" /> is Kripke-Platek set theory, whose universe is an admissible set containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" />.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {W-KPI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {W-KPI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef52d404fe0721f1485250c540e246548fda9dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.779ex; height:2.343ex;" alt="{\displaystyle {\mathsf {W-KPI}}}" /> is a weakened version of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ed2fad9bd3680aec68c9bcc7595de817ab34cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.745ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPI}}}" /> based on W-types.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPI}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ed2fad9bd3680aec68c9bcc7595de817ab34cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.745ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPI}}}" /> asserts that the universe is a limit of admissible sets.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {W-KPi}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {W-KPi}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1537b1fe3440ffcfdb16ec050284c992821998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.689ex; height:2.343ex;" alt="{\displaystyle {\mathsf {W-KPi}}}" /> is a weakened version of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830e97ea8ec435efbeea4e4005b3127c08799cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPi}}}" /> based on W-types.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPi}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPi}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830e97ea8ec435efbeea4e4005b3127c08799cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPi}}}" /> asserts that the universe is inaccessible sets.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPh}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPh}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1146ae4affa7cdac8e348380e4b7ab10a6e6b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.3ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPh}}}" /> asserts that the universe is hyperinaccessible: an inaccessible set and a limit of inaccessible sets.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPM}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbebfc6d39aa000417f7935132cfe0724795a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.132ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KPM}}}" /> asserts that the universe is a Mahlo set.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP+\Pi }}_{\mathsf {n}}-{\mathsf {Ref}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">Π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP+\Pi }}_{\mathsf {n}}-{\mathsf {Ref}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f621b693be07849d9c2ab9ca7bab05710279394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.848ex; height:2.509ex;" alt="{\displaystyle {\mathsf {KP+\Pi }}_{\mathsf {n}}-{\mathsf {Ref}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94295759f4f38c35275d7f87f61ec91690ef5028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KP}}}" /> augmented by a certain first-order reflection scheme.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Stability}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">b</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">l</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">y</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Stability}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ad48acf3b8e5c9d3596e36eb7b588569eae57e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.033ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Stability}}}" /> is KPi augmented by the axiom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \alpha \exists \kappa \geq \alpha (L_{\kappa }\preceq _{1}L_{\kappa +\alpha })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mi mathvariant="normal">∃</mi> <mi>κ</mi> <mo>≥</mo> <mi>α</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> </msub> <msub> <mo>⪯</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> <mo>+</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \alpha \exists \kappa \geq \alpha (L_{\kappa }\preceq _{1}L_{\kappa +\alpha })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e61fb28aaedaae7ff78e9cfa6a35cfd73a7d63f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.814ex; height:2.843ex;" alt="{\displaystyle \forall \alpha \exists \kappa \geq \alpha (L_{\kappa }\preceq _{1}L_{\kappa +\alpha })}" />.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KPM}}^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KPM}}^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c6f2cff8c5e90582fda564ac6c248b0806be27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.643ex; height:2.509ex;" alt="{\displaystyle {\mathsf {KPM}}^{+}}" /> is KPI augmented by the assertion "at least one recursively Mahlo ordinal exists".</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}\omega +(M\prec _{\Sigma _{1}}V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mi>ω</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>M</mi> <msub> <mo>≺</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}\omega +(M\prec _{\Sigma _{1}}V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0592e7218df0a9127e701c3d1e641507141f000e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.772ex; height:3.009ex;" alt="{\displaystyle {\mathsf {KP}}\omega +(M\prec _{\Sigma _{1}}V)}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {KP}}\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">K</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {KP}}\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a04b445af4afc4d37b59016b716bccc85a4362c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.544ex; height:2.176ex;" alt="{\displaystyle {\mathsf {KP}}\omega }" /> with an axiom stating that 'there exists a non-empty and transitive set M such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\prec _{\Sigma _{1}}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msub> <mo>≺</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\prec _{\Sigma _{1}}V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81474177b319691b55e465f82dd32164af4a327d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.578ex; height:2.843ex;" alt="{\displaystyle M\prec _{\Sigma _{1}}V}" />'.</li></ul></li></ul> A superscript zero indicates that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \in }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:1.843ex;" alt="{\displaystyle \in }" />-induction is removed (making the theory significantly weaker). <ul><li>Type theory <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CPRC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CPRC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9fc5f80a0e1166f073df28e390b969514f8cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.957ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CPRC}}}" /> is the Herbelin-Patey Calculus of Primitive Recursive Constructions.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{\mathsf {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{\mathsf {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de7a71052f61d1c022520017e87fa8b7fc1ef626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.376ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{\mathsf {n}}}" /> is type theory without W-types and with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> universes.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{<\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{<\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82935c3041767bfed1b2bb3c5c2a58da30d7c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.827ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{<\omega }}" /> is type theory without W-types and with finitely many universes.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {MLU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {MLU}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b814b30872af233e4e5a5656b602d181e7119b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.893ex; height:2.176ex;" alt="{\displaystyle {\mathsf {MLU}}}" /> is type theory with a next universe operator.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {MLS}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {MLS}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a8c0a7c6d456061ae1ef8238669819e93cbfbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.586ex; height:2.176ex;" alt="{\displaystyle {\mathsf {MLS}}}" /> is type theory without W-types and with a superuniverse.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(ML)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(ML)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eaadfb21c83d4b1bd3a3193cc665c086de66907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.695ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(ML)}}}" /> is an automorphism on type theory without W-types.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{1}{\mathsf {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c99d9d51817bf51d607f93796feeaee0d16e229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.898ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {V}}}" /> is type theory with one universe and Aczel's type of iterative sets.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {MLW}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {MLW}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9d47ecc043b2f225c300799af4e0cc25e38997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.487ex; height:2.176ex;" alt="{\displaystyle {\mathsf {MLW}}}" /> is type theory with indexed W-Types.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{1}{\mathsf {W}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed399b8e22e64a92f8ab8b3d8e7be24ae2d20ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.541ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{1}{\mathsf {W}}}" /> is type theory with W-types and one universe.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ML}}_{<\omega }{\mathsf {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>ω</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {ML}}_{<\omega }{\mathsf {W}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071d3bb9b4492129e3cdfedae8ff1ccef298b42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.02ex; height:2.509ex;" alt="{\displaystyle {\mathsf {ML}}_{<\omega }{\mathsf {W}}}" /> is type theory with W-types and finitely many universes.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(MLW)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">L</mi> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(MLW)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233474cc9bd8b8ee7a3bcda02553dad198993259" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.888ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(MLW)}}}" /> is an automorphism on type theory with W-types.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {TTM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {TTM}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d18184b98126bd325a8d7485a8c421f10260dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.199ex; height:2.176ex;" alt="{\displaystyle {\mathsf {TTM}}}" /> is type theory with a Mahlo universe.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38b84807ebde16d0af8fc1ffa9500957b272bd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.518ex; height:2.176ex;" alt="{\displaystyle \lambda 2}" /> is <a href="/wiki/System_F" title="System F">System F</a>, also polymorphic lambda calculus or second-order lambda calculus.</li></ul></li></ul> <ul><li>Constructive set theory <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68735fcc4a56810ef5e5662f4ab1a4ad4dde9e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CZF}}}" /> is Aczel's constructive set theory.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF+REA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF+REA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca379dbbbec7fb23f6550147a48e72262c3aca38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.508ex; height:2.343ex;" alt="{\displaystyle {\mathsf {CZF+REA}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68735fcc4a56810ef5e5662f4ab1a4ad4dde9e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CZF}}}" /> plus the regular extension axiom.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF+REA+FZ}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">A</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">Z</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF+REA+FZ}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486a9679895c850052e5d3762c04da92bbcf5c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.146ex; height:2.509ex;" alt="{\displaystyle {\mathsf {CZF+REA+FZ}}_{2}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF+REA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF+REA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca379dbbbec7fb23f6550147a48e72262c3aca38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.508ex; height:2.343ex;" alt="{\displaystyle {\mathsf {CZF+REA}}}" /> plus the full-second order induction scheme.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZFM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZFM}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e084eeb1df10cdae337b468508c7c8d3846e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.262ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CZFM}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CZF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CZF}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68735fcc4a56810ef5e5662f4ab1a4ad4dde9e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CZF}}}" /> with a Mahlo universe.</li></ul></li></ul> <ul><li>Explicit mathematics <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ee7cbff02ef448768cfae28673e1020c89e9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.475ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}}" /> is basic explicit mathematics plus elementary comprehension</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+JR}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">J</mi> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+JR}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87fe56509e950873e584351c442292997f9ddc40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.883ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+JR}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ee7cbff02ef448768cfae28673e1020c89e9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.475ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}}" /> plus join rule</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">J</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98dda0d506192947f139e8e3179e39c6274e2b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.381ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ee7cbff02ef448768cfae28673e1020c89e9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.475ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}}" /> plus join axioms</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EON}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">O</mi> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EON}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8de374758b8d4d7868f5387ee49c3539cc5d8ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.744ex; height:2.176ex;" alt="{\displaystyle {\mathsf {EON}}}" /> is a weak variant of the <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Feferman</a>'s <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ab81b830d372ce7c5d1be9545f4710a34a0279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle {\mathsf {T}}_{0}}" />.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ab81b830d372ce7c5d1be9545f4710a34a0279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle {\mathsf {T}}_{0}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J+IG}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">J</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J+IG}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9dce5933973353feda8ace12a562626053bdf29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.419ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J+IG}}}" />, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {IG}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {IG}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8cd9224a6c475df5331045cd1369eafc9df9774" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle {\mathsf {IG}}}" /> is inductive generation.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7860e4d66c67ce669b88f07a796b143537193daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle {\mathsf {T}}}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J+IG+FZ}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">J</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">G</mi> <mo mathvariant="sans-serif">+</mo> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">Z</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J+IG+FZ}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83787d0eeed8c49749d33956fc21a4f87c69b72a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.056ex; height:2.509ex;" alt="{\displaystyle {\mathsf {EM}}_{0}{\mathsf {+J+IG+FZ}}_{2}}" />, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {FZ}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">Z</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {FZ}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918c7dea171752ed405d9cda67398d7d9cc670b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.797ex; height:2.509ex;" alt="{\displaystyle {\mathsf {FZ}}_{2}}" /> is the full second-order induction scheme.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=17" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li> <li><a href="/wiki/Large_cardinal_property" class="mw-redirect" title="Large cardinal property">Large cardinal property</a></li> <li><a href="/wiki/Feferman%E2%80%93Sch%C3%BCtte_ordinal" title="Feferman–Schütte ordinal">Feferman–Schütte ordinal</a></li> <li><a href="/wiki/Bachmann%E2%80%93Howard_ordinal" title="Bachmann–Howard ordinal">Bachmann–Howard ordinal</a></li> <li><a href="/wiki/Complexity_class" title="Complexity class">Complexity class</a></li> <li><a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">Gentzen's consistency proof</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=18" title="Edit section: Notes">edit</a>]</div> <dl><dd>1.<style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><a href="#ref_a">^</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<n\leq {\mathsf {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>n</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<n\leq {\mathsf {\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ddb94b4758f4297703d31b81f2ed95cdacc844f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.2ex; height:2.343ex;" alt="{\displaystyle 1<n\leq {\mathsf {\omega }}}" /></dd> <dd>2.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_b">^</a> The Veblen function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /> with countably infinitely iterated least fixed points.[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>]</dd> <dd>3.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_c">^</a> Can also be commonly written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\varepsilon _{\Omega +1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\varepsilon _{\Omega +1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b5858299158dec1d30f5caadb2629d9a966d547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle \psi (\varepsilon _{\Omega +1})}" /> in Madore's ψ.</dd> <dd>4.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_d">^</a> Uses Madore's ψ rather than Buchholz's ψ.</dd> <dd>5.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_e">^</a> Can also be commonly written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\varepsilon _{\Omega _{\omega }+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\varepsilon _{\Omega _{\omega }+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0199d0e2ddcc6347dc0561a65d46a4034d6add96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.92ex; height:2.843ex;" alt="{\displaystyle \psi (\varepsilon _{\Omega _{\omega }+1})}" /> in Madore's ψ.</dd> <dd>6.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_f">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /> represents the first recursively weakly compact ordinal. Uses Arai's ψ rather than Buchholz's ψ.</dd> <dd>7.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_g">^</a> Also the proof-theoretic ordinal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {Aut(W-ID)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> <mo mathvariant="sans-serif" stretchy="false">(</mo> <mi mathvariant="sans-serif">W</mi> <mo mathvariant="sans-serif">−</mo> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">D</mi> <mo mathvariant="sans-serif" stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Aut(W-ID)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c68798fc3699c7268c6c0fa606c2286c2dcb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.76ex; height:2.843ex;" alt="{\displaystyle {\mathsf {Aut(W-ID)}}}" />, as the amount of weakening given by the W-types is not enough.</dd> <dd>8.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_h">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" /> represents the first inaccessible cardinal. Uses Jäger's ψ rather than Buchholz's ψ.</dd> <dd>9.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_i">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /> represents the limit of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" />-inaccessible cardinals. Uses (presumably) Jäger's ψ.</dd> <dd>10.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_j">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a3547ba2f3cc5cb4463473815f227092b4766a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.343ex;" alt="{\displaystyle L^{*}}" />represents the limit of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" />-inaccessible cardinals. Uses (presumably) Jäger's ψ.</dd> <dd>11.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_k">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> represents the first Mahlo cardinal. Uses Rathjen's ψ rather than Buchholz's ψ.</dd> <dd>12.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_l">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /> represents the first weakly compact cardinal. Uses Rathjen's Ψ rather than Buchholz's ψ.</dd> <dd>13.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_m">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fcfdbcd1348cf9e34618a31dbdcb36361406220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \Xi }" /> represents the first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{0}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{0}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2585e973168fcf1f1ab9ec70ab0e8887d6bf6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{0}^{2}}" />-indescribable cardinal. Uses Stegert's Ψ rather than Buchholz's ψ.</dd> <dd>14.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_n">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /> is the smallest <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \theta <Y\exists \kappa <Y(}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>θ</mi> <mo><</mo> <mi>Y</mi> <mi mathvariant="normal">∃</mi> <mi>κ</mi> <mo><</mo> <mi>Y</mi> <mo stretchy="false">(</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \theta <Y\exists \kappa <Y(}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf4cda64fa656afbc3a57f829a0ce1de0035c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.663ex; height:2.843ex;" alt="{\displaystyle \forall \theta <Y\exists \kappa <Y(}" />'<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" />-indescribable') and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \theta <Y\forall \kappa <Y(}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>θ</mi> <mo><</mo> <mi>Y</mi> <mi mathvariant="normal">∀</mi> <mi>κ</mi> <mo><</mo> <mi>Y</mi> <mo stretchy="false">(</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \theta <Y\forall \kappa <Y(}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ea91a5bee5740ff5bfff2b3c89fc62f79aef2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.663ex; height:2.843ex;" alt="{\displaystyle \forall \theta <Y\forall \kappa <Y(}" />'<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" />-indescribable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow \theta <\kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> <mi>θ</mi> <mo><</mo> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow \theta <\kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c01dd28cb5e5cf3a228d31843c57c0064252b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.497ex; height:2.176ex;" alt="{\displaystyle \rightarrow \theta <\kappa }" />'). Uses Stegert's Ψ rather than Buchholz's ψ.</dd> <dd>15.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><a href="#ref_o">^</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> represents the first Mahlo cardinal. Uses (presumably) Rathjen's ψ.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Ordinal_analysis&action=edit&section=19" title="Edit section: Citations">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> M. Rathjen, "<a rel="nofollow" class="external text" href="https://www1.maths.leeds.ac.uk/~rathjen/BEYOND.pdf">Admissible Proof Theory and Beyond</a>". In Studies in Logic and the Foundations of Mathematics vol. 134 (1995), pp.123--147. </li> <li id="cite_note-Realm-2">^ <a href="#cite_ref-Realm_2-0">a</a> <a href="#cite_ref-Realm_2-1">b</a> <a href="#cite_ref-Realm_2-2">c</a> Rathjen, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231207051454/http://www1.maths.leeds.ac.uk/~rathjen/realm.pdf">The Realm of Ordinal Analysis</a>. Accessed 2021 September 29. </li> <li id="cite_note-Krajicek-3"><a href="#cite_ref-Krajicek_3-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKrajicek1995" class="citation book cs1">Krajicek, Jan (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/boundedarithmeti0000kraj/page/18">Bounded Arithmetic, Propositional Logic and Complexity Theory</a>. 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"An ordinal analysis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f925c39d0aeb2bf320ab2adcc5f3608e672465fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle \Pi _{N}}" />-Collection". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2311.12459">2311.12459</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=An+ordinal+analysis+of+MATH+RENDER+ERROR-Collection&rft.date=2024-02-12&rft_id=info%3Aarxiv%2F2311.12459&rft.aulast=Arai&rft.aufirst=Toshiyasu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Valentin Blot. 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(1981), <a rel="nofollow" class="external text" href="https://archive.org/details/iteratedinductiv0000unse">Iterated inductive definitions and sub-systems of analysis</a>, Lecture Notes in Math., vol. 897, Berlin: Springer-Verlag, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0091894">10.1007/BFb0091894</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-11170-2" title="Special:BookSources/978-3-540-11170-2"><bdi>978-3-540-11170-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Iterated+inductive+definitions+and+sub-systems+of+analysis&rft.place=Berlin&rft.series=Lecture+Notes+in+Math.&rft.pub=Springer-Verlag&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2FBFb0091894&rft.isbn=978-3-540-11170-2&rft.aulast=Buchholz&rft.aufirst=W.&rft.au=Feferman%2C+S.&rft.au=Pohlers%2C+W.&rft.au=Sieg%2C+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fiteratedinductiv0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPohlers1989" class="citation cs2">Pohlers, Wolfram (1989), <a rel="nofollow" class="external text" href="https://archive.org/details/prooftheoryintro0000pohl">Proof Theory</a>, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-46825-7">10.1007/978-3-540-46825-7</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-51842-8" title="Special:BookSources/3-540-51842-8"><bdi>3-540-51842-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1026933">1026933</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proof+Theory&rft.place=Berlin&rft.series=Lecture+Notes+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1989&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1026933%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-540-46825-7&rft.isbn=3-540-51842-8&rft.aulast=Pohlers&rft.aufirst=Wolfram&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprooftheoryintro0000pohl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPohlers1998" class="citation cs2">Pohlers, Wolfram (1998), "Set Theory and Second Order Number Theory", Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 137, Amsterdam: Elsevier Science B. 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Logic, 29 (4): 249–263, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01651328">10.1007/BF01651328</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1062729">1062729</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14125063">14125063</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Arch.+Math.+Logic&rft.atitle=Ordinal+notations+based+on+a+weakly+Mahlo+cardinal.&rft.volume=29&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E249-%3C%2Fspan%3E263&rft.date=1990&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1062729%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14125063%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01651328&rft.aulast=Rathjen&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRathjen2006" class="citation cs2">Rathjen, Michael (2006), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091222002129/http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf">"The art of ordinal analysis"</a> (PDF), International Congress of Mathematicians, vol. II, Zürich: Eur. 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Soc., pp. 45–69, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2275588">2275588</a>, archived from the original on 2009-12-22</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+art+of+ordinal+analysis&rft.btitle=International+Congress+of+Mathematicians&rft.place=Z%C3%BCrich&rft.pages=%3Cspan+class%3D%22nowrap%22%3E45-%3C%2Fspan%3E69&rft.pub=Eur.+Math.+Soc.&rft.date=2006&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2275588%23id-name%3DMR&rft.aulast=Rathjen&rft.aufirst=Michael&rft_id=http%3A%2F%2Fwww.icm2006.org%2Fproceedings%2FVol_II%2Fcontents%2FICM_Vol_2_03.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRose1984" class="citation cs2">Rose, H.E. (1984), Subrecursion. Functions and Hierarchies, Oxford logic guides, vol. 9, Oxford, New York: Clarendon Press, Oxford University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Subrecursion.+Functions+and+Hierarchies&rft.place=Oxford%2C+New+York&rft.series=Oxford+logic+guides&rft.pub=Clarendon+Press%2C+Oxford+University+Press&rft.date=1984&rft.aulast=Rose&rft.aufirst=H.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchütte1977" class="citation cs2">Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-07911-4" title="Special:BookSources/3-540-07911-4"><bdi>3-540-07911-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0505313">0505313</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proof+theory&rft.place=Berlin-New+York&rft.series=Grundlehren+der+Mathematischen+Wissenschaften&rft.pages=xii%2B299&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=3-540-07911-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0505313%23id-name%3DMR&rft.aulast=Sch%C3%BCtte&rft.aufirst=Kurt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSetzer2004" class="citation cs2">Setzer, Anton (2004), <a rel="nofollow" class="external text" href="https://journals.openedition.org/msh/pdf/2959">"Proof theory of Martin-Löf type theory. An Overview"</a>, Mathématiques et Sciences Humaines. Mathematics and Social Sciences (165): 59–99</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math%C3%A9matiques+et+Sciences+Humaines.+Mathematics+and+Social+Sciences&rft.atitle=Proof+theory+of+Martin-L%C3%B6f+type+theory.+An+Overview&rft.issue=165&rft.pages=%3Cspan+class%3D%22nowrap%22%3E59-%3C%2Fspan%3E99&rft.date=2004&rft.aulast=Setzer&rft.aufirst=Anton&rft_id=https%3A%2F%2Fjournals.openedition.org%2Fmsh%2Fpdf%2F2959&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTakeuti1987" class="citation cs2">Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics, vol. 81 (Second ed.), Amsterdam: North-Holland Publishing Co., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-87943-9" title="Special:BookSources/0-444-87943-9"><bdi>0-444-87943-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0882549">0882549</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proof+theory&rft.place=Amsterdam&rft.series=Studies+in+Logic+and+the+Foundations+of+Mathematics&rft.edition=Second&rft.pub=North-Holland+Publishing+Co.&rft.date=1987&rft.isbn=0-444-87943-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0882549%23id-name%3DMR&rft.aulast=Takeuti&rft.aufirst=Gaisi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRathjen1994" class="citation cs2">Rathjen, Michael (1994), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/0168-0072%2894%2990074-4">"Proof theory of Reflection"</a>, Annals of Pure and Applied Logic, 68 (2): 181–224, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0168-0072%2894%2990074-4">10.1016/0168-0072(94)90074-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Pure+and+Applied+Logic&rft.atitle=Proof+theory+of+Reflection&rft.volume=68&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E181-%3C%2Fspan%3E224&rft.date=1994&rft_id=info%3Adoi%2F10.1016%2F0168-0072%2894%2990074-4&rft.aulast=Rathjen&rft.aufirst=Michael&rft_id=https%3A%2F%2Fdx.doi.org%2F10.1016%2F0168-0072%252894%252990074-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStegert2010" class="citation cs2">Stegert, Jan-Carl (2010), <a rel="nofollow" class="external text" href="https://d-nb.info/1017849250/34">Ordinal Proof Theory of Kripke-Platek Set Theory Augmented by Strong Reflection Principles</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinal+Proof+Theory+of+Kripke-Platek+Set+Theory+Augmented+by+Strong+Reflection+Principles&rft.date=2010&rft.aulast=Stegert&rft.aufirst=Jan-Carl&rft_id=https%3A%2F%2Fd-nb.info%2F1017849250%2F34&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+analysis" class="Z3988"></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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</div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-76d4c66f66-pcgkf","wgBackendResponseTime":355,"wgPageParseReport":{"limitreport":{"cputime":"1.071","walltime":"1.337","ppvisitednodes":{"value":11552,"limit":1000000},"postexpandincludesize":{"value":79436,"limit":2097152},"templateargumentsize":{"value":3691,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":129284,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 734.981 1 -total"," 23.34% 171.508 1 Template:Reflist"," 18.42% 135.347 24 Template:Rp"," 17.28% 127.021 24 Template:R/superscript"," 14.91% 109.571 1 Template:Short_description"," 12.40% 91.123 2 Template:Cite_book"," 10.93% 80.326 2 Template:Pagetype"," 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Ordinal analysis - Wikipedia