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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p> The <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> statements discussed below are provably <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">independent</a> of <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a> (the canonical <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> of contemporary mathematics, consisting of the <a href="/wiki/Zermelo%E2%80%93Fraenkel_axioms" class="mw-redirect" title="Zermelo–Fraenkel axioms">Zermelo–Fraenkel axioms</a> plus the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>), assuming that ZFC is <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a>. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Axiomatic_set_theory">Axiomatic set theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=1" title="Edit section: Axiomatic set theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">Axiomatic set theory</a></div> <p>In 1931, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> proved his <a href="/wiki/G%C3%B6del%27s_second_incompleteness_theorem" class="mw-redirect" title="Gödel's second incompleteness theorem">incompleteness theorems</a>, establishing that many mathematical theories, including ZFC, cannot prove their own consistency. Assuming <a href="/wiki/%CE%A9-consistency" class="mw-redirect" title="Ω-consistency">ω-consistency</a> of such a theory, the consistency statement can also not be disproven, meaning it is independent. A few years later, other arithmetic statements were defined that are independent of any such theory, see for example <a href="/wiki/Rosser%27s_trick" title="Rosser's trick">Rosser's trick</a>. </p><p>The following set theoretic statements are independent of ZFC, among others: </p> <ul><li>the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; <a href="/wiki/Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Paul Cohen</a> later invented the method of <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a> to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four independence results are also due to Gödel/Cohen.);</li> <li>the <a href="/wiki/Generalized_continuum_hypothesis" class="mw-redirect" title="Generalized continuum hypothesis">generalized continuum hypothesis</a> (GCH);</li> <li>a related independent statement is that if a set <i>x</i> has fewer elements than <i>y</i>, then <i>x</i> also has fewer <a href="/wiki/Subset" title="Subset">subsets</a> than <i>y</i>. In particular, this statement fails when the cardinalities of the power sets of <i>x</i> and <i>y</i> coincide;</li> <li>the <a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">axiom of constructibility</a> (<i>V</i> = <i>L</i>);</li> <li>the <a href="/wiki/Diamond_principle" title="Diamond principle">diamond principle</a> (◊);</li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a> (MA);</li> <li>MA + ¬CH (independence shown by <a href="/wiki/Robert_Solovay" class="mw-redirect" title="Robert Solovay">Solovay</a> and <a href="/wiki/Stanley_Tennenbaum" title="Stanley Tennenbaum">Tennenbaum</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>Every <a href="/wiki/Aronszajn_tree" title="Aronszajn tree">Aronszajn tree</a> is special (EATS);</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Implication_Chains_of_Undecidable_ZFC_Statements.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Implication_Chains_of_Undecidable_ZFC_Statements.png/220px-Implication_Chains_of_Undecidable_ZFC_Statements.png" decoding="async" width="220" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Implication_Chains_of_Undecidable_ZFC_Statements.png/330px-Implication_Chains_of_Undecidable_ZFC_Statements.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Implication_Chains_of_Undecidable_ZFC_Statements.png/440px-Implication_Chains_of_Undecidable_ZFC_Statements.png 2x" data-file-width="502" data-file-height="654" /></a><figcaption>Diagram showing the implication chains</figcaption></figure> <p>We have the following chains of implications: </p> <dl><dd><i>V</i> = <i>L</i> → ◊ → CH,</dd> <dd><i>V</i> = <i>L</i> → GCH → CH,</dd> <dd>CH → MA,</dd></dl> <p>and (see section on order theory): </p> <dl><dd>◊ → ¬<a href="/wiki/Suslin_hypothesis" class="mw-redirect" title="Suslin hypothesis">SH</a>,</dd> <dd>MA + ¬CH → EATS → SH.</dd></dl> <p>Several statements related to the existence of <a href="/wiki/Large_cardinal" title="Large cardinal">large cardinals</a> cannot be proven in ZFC (assuming ZFC is consistent). These are independent of ZFC provided that they are consistent with ZFC, which most working set theorists believe to be the case. These statements are strong enough to imply the consistency of ZFC. This has the consequence (via <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's second incompleteness theorem</a>) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent). The following statements belong to this class: </p> <ul><li>Existence of <a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible cardinals</a></li> <li>Existence of <a href="/wiki/Mahlo_cardinal" title="Mahlo cardinal">Mahlo cardinals</a></li> <li>Existence of <a href="/wiki/Measurable_cardinal" title="Measurable cardinal">measurable cardinals</a> (first conjectured by <a href="/wiki/Stanislaw_Ulam" class="mw-redirect" title="Stanislaw Ulam">Ulam</a>)</li> <li>Existence of <a href="/wiki/Supercompact_cardinal" title="Supercompact cardinal">supercompact cardinals</a></li></ul> <p>The following statements can be proven to be independent of ZFC assuming the consistency of a suitable large cardinal: </p> <ul><li><a href="/wiki/Proper_forcing_axiom" title="Proper forcing axiom">Proper forcing axiom</a></li> <li><a href="/wiki/Open_coloring_axiom" title="Open coloring axiom">Open coloring axiom</a></li> <li><a href="/wiki/Martin%27s_maximum" title="Martin's maximum">Martin's maximum</a></li> <li>Existence of <a href="/wiki/Zero_sharp" title="Zero sharp">0<sup>#</sup></a></li> <li><a href="/wiki/Singular_cardinals_hypothesis" title="Singular cardinals hypothesis">Singular cardinals hypothesis</a></li> <li><a href="/wiki/Projective_determinacy" class="mw-redirect" title="Projective determinacy">Projective determinacy</a> (and even the full <a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">axiom of determinacy</a> if the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> is not assumed)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Set_theory_of_the_real_line">Set theory of the real line</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=2" title="Edit section: Set theory of the real line"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Set_theory_of_the_real_line" title="Set theory of the real line">Set theory of the real line</a></div> <p>There are many <a href="/wiki/Cardinal_invariant" class="mw-redirect" title="Cardinal invariant">cardinal invariants</a> of the real line, connected with <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> and statements related to the <a href="/wiki/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a>, whose exact values are independent of ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any <a href="/wiki/Regular_cardinal" title="Regular cardinal">regular cardinal</a> between <a href="/wiki/Aleph_number#Aleph-one" title="Aleph number">ℵ<sub>1</sub></a> and <a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">2<sup>ℵ<sub>0</sub></sup></a>. This is a major area of study in the set theory of the real line (see <a href="/wiki/Cichon_diagram" class="mw-redirect" title="Cichon diagram">Cichon diagram</a>). MA has a tendency to set most interesting cardinal invariants equal to 2<sup>ℵ<sub>0</sub></sup>. </p><p>A subset <i>X</i> of the real line is a <a href="/wiki/Strong_measure_zero_set" title="Strong measure zero set">strong measure zero set</a> if to every sequence (<i>ε<sub>n</sub></i>) of positive reals there exists a sequence of intervals (<i>I<sub>n</sub></i>) which covers <i>X</i> and such that <i>I<sub>n</sub></i> has length at most <i>ε<sub>n</sub></i>. Borel's conjecture, that every strong measure zero set is countable, is independent of ZFC. </p><p>A subset <i>X</i> of the real line is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>-dense if every open interval contains <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>-many elements of <i>X</i>. Whether all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>-dense sets are order-isomorphic is independent of ZFC.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Order_theory">Order theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=3" title="Edit section: Order theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Order_theory" title="Order theory">Order theory</a></div> <p><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a> asks whether a specific short list of properties characterizes the ordered set of real numbers <b>R</b>. This is undecidable in ZFC.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> A <i>Suslin line</i> is an ordered set which satisfies this specific list of properties but is not order-isomorphic to <b>R</b>. The <a href="/wiki/Diamond_principle" title="Diamond principle">diamond principle</a> ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS (<a href="/wiki/Every_Aronszajn_tree_is_special" class="mw-redirect" title="Every Aronszajn tree is special">every Aronszajn tree is special</a>),<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> which in turn implies (but is not equivalent to)<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the nonexistence of Suslin lines. <a href="/wiki/Ronald_Jensen" title="Ronald Jensen">Ronald Jensen</a> proved that CH does not imply the existence of a Suslin line.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Existence of <a href="/wiki/Kurepa_tree" title="Kurepa tree">Kurepa trees</a> is independent of ZFC, assuming consistency of an <a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible cardinal</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Existence of a partition of the <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b914a8bfef5d1b9b106048afa0aab4a99251f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}"></span> into two colors with no monochromatic uncountable sequentially closed subset is independent of ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of a <a href="/wiki/Mahlo_cardinal" title="Mahlo cardinal">Mahlo cardinal</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This theorem of <a href="/wiki/Saharon_Shelah" title="Saharon Shelah">Shelah</a> answers a question of <a href="/wiki/Harvey_Friedman" title="Harvey Friedman">H. Friedman</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Abstract_algebra">Abstract algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=4" title="Edit section: Abstract algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></div> <p>In 1973, <a href="/wiki/Saharon_Shelah" title="Saharon Shelah">Saharon Shelah</a> showed that the <a href="/wiki/Whitehead_problem" title="Whitehead problem">Whitehead problem</a> ("is every <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> <i>A</i> with <a href="/wiki/Ext_functor" title="Ext functor">Ext</a><sup>1</sup>(A, <b>Z</b>) = 0 a <a href="/wiki/Free_abelian_group" title="Free abelian group">free abelian group</a>?") is independent of ZFC.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> An abelian group with Ext<sup>1</sup>(A, <b>Z</b>) = 0 is called a Whitehead group; <a href="/wiki/Martin%27s_Axiom" class="mw-redirect" title="Martin's Axiom">MA</a> + ¬CH proves the existence of a non-free Whitehead group, while <i>V</i> = <i>L</i> proves that all Whitehead groups are free. In one of the earliest applications of proper <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a>, Shelah constructed a model of ZFC + CH in which there is a non-free Whitehead group.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Consider the ring <i>A</i> = <b>R</b>[<i>x</i>,<i>y</i>,<i>z</i>] of polynomials in three variables over the real numbers and its <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> <i>M</i> = <b>R</b>(<i>x</i>,<i>y</i>,<i>z</i>). The <a href="/wiki/Projective_dimension" class="mw-redirect" title="Projective dimension">projective dimension</a> of <i>M</i> as <i>A</i>-module is either 2 or 3, but it is independent of ZFC whether it is equal to 2; it is equal to 2 if and only if CH holds.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Direct_product" title="Direct product">direct product</a> of countably many <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> has <a href="/wiki/Global_dimension" title="Global dimension">global dimension</a> 2 if and only if the continuum hypothesis holds.<sup id="cite_ref-oso_15-0" class="reference"><a href="#cite_note-oso-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Number_theory">Number theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=5" title="Edit section: Number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Number_theory" title="Number theory">Number theory</a></div> <p>One can write down a concrete polynomial <i>p</i> ∈ <b>Z</b>[<i>x</i><sub>1</sub>, ..., <i>x</i><sub>9</sub>] such that the statement "there are integers <i>m</i><sub>1</sub>, ..., <i>m</i><sub>9</sub> with <i>p</i>(<i>m</i><sub>1</sub>, ..., <i>m</i><sub>9</sub>) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). This follows from <a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a>'s resolution of <a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>; the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Measure_theory">Measure theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=6" title="Edit section: Measure theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></div> <p>A stronger version of <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a> for positive functions, where the function is no longer assumed to be <a href="/wiki/Measurable" class="mw-redirect" title="Measurable">measurable</a> but merely that the two iterated integrals are well defined and exist, is independent of ZFC. On the one hand, CH implies that there exists a function on the unit square whose iterated integrals are not equal — the function is simply the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of an ordering of [0, 1] equivalent to a <a href="/wiki/Well_ordering" class="mw-redirect" title="Well ordering">well ordering</a> of the cardinal ω<sub>1</sub>. A similar example can be constructed using <a href="/wiki/Martin%27s_axiom" title="Martin's axiom">MA</a>. On the other hand, the consistency of the strong Fubini theorem was first shown by <a href="/wiki/Harvey_Friedman" title="Harvey Friedman">Friedman</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> It can also be deduced from a variant of <a href="/wiki/Freiling%27s_axiom_of_symmetry" title="Freiling's axiom of symmetry">Freiling's axiom of symmetry</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Topology">Topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=7" title="Edit section: Topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Topology" title="Topology">Topology</a></div> <p>The Normal Moore Space conjecture, namely that every <a href="/wiki/Normal_space" title="Normal space">normal</a> <a href="/wiki/Moore_space_(topology)" title="Moore space (topology)">Moore space</a> is <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable</a>, can be disproven assuming the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> or assuming both <a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a> and the negation of the continuum hypothesis, and can be proven assuming a certain axiom which implies the existence of <a href="/wiki/Large_cardinal" title="Large cardinal">large cardinals</a>. Thus, granted large cardinals, the Normal Moore Space conjecture is independent of ZFC.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>The existence of an <a href="/wiki/S-space" class="mw-redirect" title="S-space">S-space</a> is independent of ZFC. In particular, it is implied by the existence of a Suslin line.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Functional_analysis">Functional analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=8" title="Edit section: Functional analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></div> <p><a href="/w/index.php?title=Garth_Dales&action=edit&redlink=1" class="new" title="Garth Dales (page does not exist)">Garth Dales</a> and <a href="/wiki/Robert_M._Solovay" title="Robert M. Solovay">Robert M. Solovay</a> proved in 1976 that <a href="/wiki/Kaplansky%27s_conjecture" class="mw-redirect" title="Kaplansky's conjecture">Kaplansky's conjecture</a>, namely that every <a href="/wiki/Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> from the <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a> <i>C(X)</i> (where <i>X</i> is some <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a>) into any other Banach algebra must be continuous, is independent of ZFC. CH implies that for any infinite <i>X</i> there exists a discontinuous homomorphism into any Banach algebra.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Consider the algebra <i>B</i>(<i>H</i>) of <a href="/wiki/Bounded_operator" title="Bounded operator">bounded linear operators</a> on the infinite-dimensional <a href="/wiki/Hilbert_space#Separable_spaces" title="Hilbert space">separable</a> <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <i>H</i>. The <a href="/wiki/Compact_operator" title="Compact operator">compact operators</a> form a two-sided ideal in <i>B</i>(<i>H</i>). The question of whether this ideal is the sum of two properly smaller ideals is independent of ZFC, as was proved by <a href="/wiki/Andreas_Blass" title="Andreas Blass">Andreas Blass</a> and <a href="/wiki/Saharon_Shelah" title="Saharon Shelah">Saharon Shelah</a> in 1987.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p><a href="/w/index.php?title=Charles_Akemann&action=edit&redlink=1" class="new" title="Charles Akemann (page does not exist)">Charles Akemann</a> and <a href="/w/index.php?title=Nik_Weaver&action=edit&redlink=1" class="new" title="Nik Weaver (page does not exist)">Nik Weaver</a> showed in 2003 that the statement "there exists a counterexample to <a href="/wiki/Naimark%27s_problem" title="Naimark's problem">Naimark's problem</a> which is generated by ℵ<sub>1</sub>, elements" is independent of ZFC. </p><p>Miroslav Bačák and <a href="/w/index.php?title=Petr_H%C3%A1jek_(mathematician,_functional_analysis)&action=edit&redlink=1" class="new" title="Petr Hájek (mathematician, functional analysis) (page does not exist)">Petr Hájek</a> proved in 2008 that the statement "every <a href="/wiki/Asplund_space" title="Asplund space">Asplund space</a> of density character ω<sub>1</sub> has a renorming with the <a href="/w/index.php?title=Mazur_intersection_property&action=edit&redlink=1" class="new" title="Mazur intersection property (page does not exist)">Mazur intersection property</a>" is independent of ZFC. The result is shown using <a href="/wiki/Martin%27s_maximum" title="Martin's maximum">Martin's maximum</a> axiom, while Mar Jiménez and José Pedro Moreno (1997) had presented a counterexample assuming CH. </p><p>As shown by <a href="/wiki/Ilijas_Farah" title="Ilijas Farah">Ilijas Farah</a><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> and <a href="/w/index.php?title=N._Christopher_Phillips&action=edit&redlink=1" class="new" title="N. Christopher Phillips (page does not exist)">N. Christopher Phillips</a> and <a href="/w/index.php?title=Nik_Weaver&action=edit&redlink=1" class="new" title="Nik Weaver (page does not exist)">Nik Weaver</a>,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> the existence of outer automorphisms of the <a href="/wiki/Calkin_algebra" title="Calkin algebra">Calkin algebra</a> depends on set theoretic assumptions beyond ZFC. </p><p><a href="/wiki/Wetzel%27s_problem" title="Wetzel's problem">Wetzel's problem</a>, which asks if every set of <a href="/wiki/Analytic_functions" class="mw-redirect" title="Analytic functions">analytic functions</a> which takes at most countably many distinct values at every point is necessarily countable, is true if and only if the continuum hypothesis is false.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Model_theory">Model theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=9" title="Edit section: Model theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Model_theory" title="Model theory">Model theory</a></div> <p><a href="/wiki/Chang%27s_conjecture" title="Chang's conjecture">Chang's conjecture</a> is independent of ZFC assuming the consistency of an <a href="/wiki/Erd%C5%91s_cardinal" title="Erdős cardinal">Erdős cardinal</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Computability_theory">Computability theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=10" title="Edit section: Computability theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></div> <p><a href="/wiki/Marcia_Groszek" title="Marcia Groszek">Marcia Groszek</a> and <a href="/wiki/Theodore_Slaman" title="Theodore Slaman">Theodore Slaman</a> gave examples of statements independent of ZFC concerning the structure of the Turing degrees. In particular, whether there exists a maximally independent set of degrees of size less than continuum.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFKunen1980" class="citation book cs1"><a href="/wiki/Kenneth_Kunen" title="Kenneth Kunen">Kunen, Kenneth</a> (1980). <i><a href="/wiki/Set_Theory:_An_Introduction_to_Independence_Proofs" title="Set Theory: An Introduction to Independence Proofs">Set Theory: An Introduction to Independence Proofs</a></i>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-86839-9" title="Special:BookSources/0-444-86839-9"><bdi>0-444-86839-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory%3A+An+Introduction+to+Independence+Proofs&rft.pub=Elsevier&rft.date=1980&rft.isbn=0-444-86839-9&rft.aulast=Kunen&rft.aufirst=Kenneth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Baumgartner, J., All <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>-dense sets of reals can be isomorphic, Fund. Math. 79, pp.101 – 106, 1973</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolovayTennenbaum,_S.1971" class="citation journal cs1">Solovay, R. M.; Tennenbaum, S. (1971). "Iterated Cohen extensions and Souslin's problem". <i>Annals of Mathematics</i>. Second Series. <b>94</b> (2): 201–245. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970860">10.2307/1970860</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970860">1970860</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+Mathematics&rft.atitle=Iterated+Cohen+extensions+and+Souslin%27s+problem&rft.volume=94&rft.issue=2&rft.pages=201-245&rft.date=1971&rft_id=info%3Adoi%2F10.2307%2F1970860&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970860%23id-name%3DJSTOR&rft.aulast=Solovay&rft.aufirst=R.+M.&rft.au=Tennenbaum%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Baumgartner, J., J. Malitz, and W. Reiehart, Embedding trees in the rationals, Proc. Natl. Acad. Sci. U.S.A., 67, pp. 1746 – 1753, 1970</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShelah1981" class="citation journal cs1">Shelah, S. (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02762777">"Free limits of forcing and more on Aronszajn trees"</a>. <i><a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a></i>. <b>38</b> (4): 315–334. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02762777">10.1007/BF02762777</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Israel+Journal+of+Mathematics&rft.atitle=Free+limits+of+forcing+and+more+on+Aronszajn+trees&rft.volume=38&rft.issue=4&rft.pages=315-334&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2FBF02762777&rft.aulast=Shelah&rft.aufirst=S.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02762777&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Devlin, K., and H. Johnsbraten, The Souslin Problem, Lecture Notes on Mathematics 405, Springer, 1974</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Silver, J., The independence of Kurepa's conjecture and two-cardinal conjectures in model theory, in Axiomatic Set Theory, Proc. Symp, in Pure Mathematics (13) pp. 383 – 390, 1967</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Shelah, S., Proper and Improper Forcing, Springer 1992</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 – 606</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865 – 1883</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShelah1974" class="citation journal cs1">Shelah, S. (1974). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02757281">"Infinite Abelian groups, Whitehead problem and some constructions"</a>. <i><a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a></i>. <b>18</b> (3): 243–256. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02757281">10.1007/BF02757281</a></span>. <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=0357114">0357114</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Israel+Journal+of+Mathematics&rft.atitle=Infinite+Abelian+groups%2C+Whitehead+problem+and+some+constructions&rft.volume=18&rft.issue=3&rft.pages=243-256&rft.date=1974&rft_id=info%3Adoi%2F10.1007%2FBF02757281&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0357114%23id-name%3DMR&rft.aulast=Shelah&rft.aufirst=S.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02757281&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShelah1972" class="citation journal cs1">Shelah, S. (1972). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02759809">"Whitehead groups may be not free, even assuming CH, I"</a>. <i><a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a></i>. <b>28</b> (3): 193–204. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02759809">10.1007/BF02759809</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Israel+Journal+of+Mathematics&rft.atitle=Whitehead+groups+may+be+not+free%2C+even+assuming+CH%2C+I&rft.volume=28&rft.issue=3&rft.pages=193-204&rft.date=1972&rft_id=info%3Adoi%2F10.1007%2FBF02759809&rft.aulast=Shelah&rft.aufirst=S.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02759809&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShelah1980" class="citation journal cs1">Shelah, S. (1980). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02760652">"Whitehead groups may not be free even assuming CH, II"</a>. <i><a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a></i>. <b>35</b> (4): 257–285. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02760652">10.1007/BF02760652</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Israel+Journal+of+Mathematics&rft.atitle=Whitehead+groups+may+not+be+free+even+assuming+CH%2C+II&rft.volume=35&rft.issue=4&rft.pages=257-285&rft.date=1980&rft_id=info%3Adoi%2F10.1007%2FBF02760652&rft.aulast=Shelah&rft.aufirst=S.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02760652&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarbara_L._Osofsky1968" class="citation journal cs1"><a href="/wiki/Barbara_L._Osofsky" title="Barbara L. Osofsky">Barbara L. Osofsky</a> (1968). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/tran/1968-132-01/S0002-9947-1968-0224606-4/S0002-9947-1968-0224606-4.pdf">"Homological dimension and the continuum hypothesis"</a> <span class="cs1-format">(PDF)</span>. <i>Transactions of the American Mathematical Society</i>. <b>132</b>: 217–230. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1968-0224606-4">10.1090/s0002-9947-1968-0224606-4</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Homological+dimension+and+the+continuum+hypothesis&rft.volume=132&rft.pages=217-230&rft.date=1968&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-1968-0224606-4&rft.au=Barbara+L.+Osofsky&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Ftran%2F1968-132-01%2FS0002-9947-1968-0224606-4%2FS0002-9947-1968-0224606-4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-oso-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-oso_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarbara_L._Osofsky1973" class="citation book cs1"><a href="/wiki/Barbara_L._Osofsky" title="Barbara L. Osofsky">Barbara L. Osofsky</a> (1973). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rTTS6seRfSIC&q=Homological+Dimensions+of+Modules"><i>Homological Dimensions of Modules</i></a>. American Mathematical Soc. p. 60. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1662-2" title="Special:BookSources/978-0-8218-1662-2"><bdi>978-0-8218-1662-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Homological+Dimensions+of+Modules&rft.pages=60&rft.pub=American+Mathematical+Soc.&rft.date=1973&rft.isbn=978-0-8218-1662-2&rft.au=Barbara+L.+Osofsky&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrTTS6seRfSIC%26q%3DHomological%2BDimensions%2Bof%2BModules&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">See e.g.: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_P._Jones1980" class="citation journal cs1">James P. Jones (1980). <a rel="nofollow" class="external text" href="http://projecteuclid.org/download/pdf_1/euclid.bams/1183547548">"Undecidable diophantine equations"</a>. <i>Bull. Amer. Math. Soc</i>. <b>3</b> (2): 859–862. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0273-0979-1980-14832-6">10.1090/s0273-0979-1980-14832-6</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Undecidable+diophantine+equations&rft.volume=3&rft.issue=2&rft.pages=859-862&rft.date=1980&rft_id=info%3Adoi%2F10.1090%2Fs0273-0979-1980-14832-6&rft.au=James+P.+Jones&rft_id=http%3A%2F%2Fprojecteuclid.org%2Fdownload%2Fpdf_1%2Feuclid.bams%2F1183547548&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarlMoroz2014" class="citation journal cs1">Carl, M.; Moroz, B. (2014). "On a Diophantine Representation of the Predicate of Provability". <i>Journal of Mathematical Sciences</i>. <b>199</b> (199): 36–52. <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%2Fs10958-014-1830-2">10.1007/s10958-014-1830-2</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/21.11116%2F0000-0004-1E89-1">21.11116/0000-0004-1E89-1</a></span>. <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:34618563">34618563</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Sciences&rft.atitle=On+a+Diophantine+Representation+of+the+Predicate+of+Provability&rft.volume=199&rft.issue=199&rft.pages=36-52&rft.date=2014&rft_id=info%3Ahdl%2F21.11116%2F0000-0004-1E89-1&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A34618563%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10958-014-1830-2&rft.aulast=Carl&rft.aufirst=M.&rft.au=Moroz%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></li></ul> For a summary of the argument, see <a href="/wiki/Hilbert%27s_tenth_problem#Applications" title="Hilbert's tenth problem">Hilbert's tenth problem § Applications</a>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedman1980" class="citation journal cs1">Friedman, Harvey (1980). <a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1256047607">"A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions"</a>. <i>Illinois J. Math</i>. <b>24</b> (3): 390–395. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1256047607">10.1215/ijm/1256047607</a></span>. <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=0573474">0573474</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Illinois+J.+Math.&rft.atitle=A+Consistent+Fubini-Tonelli+Theorem+for+Nonmeasurable+Functions&rft.volume=24&rft.issue=3&rft.pages=390-395&rft.date=1980&rft_id=info%3Adoi%2F10.1215%2Fijm%2F1256047607&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D573474%23id-name%3DMR&rft.aulast=Friedman&rft.aufirst=Harvey&rft_id=https%3A%2F%2Fdoi.org%2F10.1215%252Fijm%252F1256047607&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreiling1986" class="citation journal cs1"><a href="/wiki/Chris_Freiling" title="Chris Freiling">Freiling, Chris</a> (1986). "Axioms of symmetry: throwing darts at the real number line". <i><a href="/wiki/Journal_of_Symbolic_Logic" title="Journal of Symbolic Logic">Journal of Symbolic Logic</a></i>. <b>51</b> (1): 190–200. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2273955">10.2307/2273955</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2273955">2273955</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=0830085">0830085</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:38174418">38174418</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Axioms+of+symmetry%3A+throwing+darts+at+the+real+number+line&rft.volume=51&rft.issue=1&rft.pages=190-200&rft.date=1986&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A38174418%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D830085%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2273955%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2273955&rft.aulast=Freiling&rft.aufirst=Chris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNyikos2001" class="citation book cs1">Nyikos, Peter J. (2001). "A history of the normal Moore space problem". <i>Handbook of the History of General Topology</i>. History of Topology. Vol. 3. Dordrecht: Kluwer Academic Publishers. pp. 1179–1212. <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%2F978-94-017-0470-0_7">10.1007/978-94-017-0470-0_7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-6970-X" title="Special:BookSources/0-7923-6970-X"><bdi>0-7923-6970-X</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=1900271">1900271</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+history+of+the+normal+Moore+space+problem&rft.btitle=Handbook+of+the+History+of+General+Topology&rft.place=Dordrecht&rft.series=History+of+Topology&rft.pages=1179-1212&rft.pub=Kluwer+Academic+Publishers&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1900271%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-94-017-0470-0_7&rft.isbn=0-7923-6970-X&rft.aulast=Nyikos&rft.aufirst=Peter+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTodorcevic1989" class="citation book cs1">Todorcevic, Stevo (1989). <i>Partition problems in topology</i>. Providence, R.I.: American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-5091-6" title="Special:BookSources/978-0-8218-5091-6"><bdi>978-0-8218-5091-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partition+problems+in+topology&rft.place=Providence%2C+R.I.&rft.pub=American+Mathematical+Society&rft.date=1989&rft.isbn=978-0-8218-5091-6&rft.aulast=Todorcevic&rft.aufirst=Stevo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFH._G._DalesW._H._Woodin1987" class="citation book cs1">H. G. Dales; W. H. Woodin (1987). <i>An introduction to independence for analysts</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+independence+for+analysts&rft.date=1987&rft.au=H.+G.+Dales&rft.au=W.+H.+Woodin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJudith_Roitman1992" class="citation journal cs1">Judith Roitman (1992). "The Uses of Set Theory". <i>Mathematical Intelligencer</i>. <b>14</b> (1).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Intelligencer&rft.atitle=The+Uses+of+Set+Theory&rft.volume=14&rft.issue=1&rft.date=1992&rft.au=Judith+Roitman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarah2011" class="citation journal cs1">Farah, Ilijas (2011). <a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2011.173.2.1">"All automorphisms of the Calkin algebra are inner"</a>. <i>Annals of Mathematics</i>. Second Series. <b>173</b> (2): 619–661. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0705.3085">0705.3085</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2011.173.2.1">10.4007/annals.2011.173.2.1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=All+automorphisms+of+the+Calkin+algebra+are+inner&rft.volume=173&rft.issue=2&rft.pages=619-661&rft.date=2011&rft_id=info%3Aarxiv%2F0705.3085&rft_id=info%3Adoi%2F10.4007%2Fannals.2011.173.2.1&rft.aulast=Farah&rft.aufirst=Ilijas&rft_id=https%3A%2F%2Fdoi.org%2F10.4007%252Fannals.2011.173.2.1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhillipsWeaver2007" class="citation journal cs1">Phillips, N. C.; Weaver, N. (2007). "The Calkin algebra has outer automorphisms". <i><a href="/wiki/Duke_Mathematical_Journal" title="Duke Mathematical Journal">Duke Mathematical Journal</a></i>. <b>139</b> (1): 185–202. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0606594">math/0606594</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-07-13915-2">10.1215/S0012-7094-07-13915-2</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:13873756">13873756</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=The+Calkin+algebra+has+outer+automorphisms&rft.volume=139&rft.issue=1&rft.pages=185-202&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0606594&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13873756%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-07-13915-2&rft.aulast=Phillips&rft.aufirst=N.+C.&rft.au=Weaver%2C+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErdős1964" class="citation journal cs1">Erdős, P. (1964). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.mmj/1028999028">"An interpolation problem associated with the continuum hypothesis"</a>. <i>The Michigan Mathematical Journal</i>. <b>11</b>: 9–10. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1307%2Fmmj%2F1028999028">10.1307/mmj/1028999028</a></span>. <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=0168482">0168482</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Michigan+Mathematical+Journal&rft.atitle=An+interpolation+problem+associated+with+the+continuum+hypothesis&rft.volume=11&rft.pages=9-10&rft.date=1964&rft_id=info%3Adoi%2F10.1307%2Fmmj%2F1028999028&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0168482%23id-name%3DMR&rft.aulast=Erd%C5%91s&rft.aufirst=P.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.mmj%2F1028999028&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span>.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGroszekSlaman1983" class="citation journal cs1"><a href="/wiki/Marcia_Groszek" title="Marcia Groszek">Groszek, Marcia J.</a>; <a href="/wiki/Theodore_Slaman" title="Theodore Slaman">Slaman, T.</a> (1983). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1999225">"Independence results on the global structure of the Turing degrees"</a>. <i><a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>. <b>277</b> (2): 579. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1999225">10.2307/1999225</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1999225">1999225</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Independence+results+on+the+global+structure+of+the+Turing+degrees&rft.volume=277&rft.issue=2&rft.pages=579&rft.date=1983&rft_id=info%3Adoi%2F10.2307%2F1999225&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1999225%23id-name%3DJSTOR&rft.aulast=Groszek&rft.aufirst=Marcia+J.&rft.au=Slaman%2C+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1999225&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+statements+independent+of+ZFC" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_statements_independent_of_ZFC&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/1924">What are some reasonable-sounding statements that are independent of ZFC?</a>, mathoverflow.net</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a class="mw-selflink selflink">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a 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