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class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a>, <a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a>, <a class="existingWikiWord" href="/nlab/show/lambda-calculus">lambda-calculus</a>, <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/simple+type+theory">simple type theory</a>, <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/collection">collection</a>, <a class="existingWikiWord" href="/nlab/show/object">object</a>, <a class="existingWikiWord" href="/nlab/show/type">type</a>, <a class="existingWikiWord" href="/nlab/show/term">term</a>, <a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a>, <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a>, <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/size+issues">size issues</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order logic</a></p> </li> </ul> <h2 id="set_theory"> Set theory</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a></strong></p> <ul> <li>fundamentals of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/propositional+logic">propositional logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/typed+predicate+logic">typed predicate logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a></li> <li><a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a>, <a class="existingWikiWord" href="/nlab/show/function">function</a>, <a class="existingWikiWord" href="/nlab/show/relation">relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/small+set">small set</a>, <a class="existingWikiWord" href="/nlab/show/large+set">large set</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/pairing+structure">pairing structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/union+structure">union structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> <li><a class="existingWikiWord" href="/nlab/show/powerset+structure">powerset structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/natural+numbers+structure">natural numbers structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> </ul> </li> <li>presentations of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+set+theory">first-order set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/unsorted+set+theory">unsorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/simply+sorted+set+theory">simply sorted set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/one-sorted+set+theory">one-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/two-sorted+set+theory">two-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/three-sorted+set+theory">three-sorted set theory</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/dependently+sorted+set+theory">dependently sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/structurally+presented+set+theory">structurally presented set theory</a></li> </ul> </li> <li>structuralism in set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a></li> <li><a class="existingWikiWord" href="/nlab/show/ZFA">ZFA</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski+set+theory">Mostowski set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/New+Foundations">New Foundations</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/categorical+set+theory">categorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a></li> <li><a class="existingWikiWord" href="/nlab/show/ETCS+with+elements">ETCS with elements</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+I">Trimble on ETCS I</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+II">Trimble on ETCS II</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+III">Trimble on ETCS III</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+ZFC">structural ZFC</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/allegorical+set+theory">allegorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/SEAR">SEAR</a></li> </ul> </li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/class-set+theory">class-set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/class">class</a>, <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+class">universal class</a>, <a class="existingWikiWord" href="/nlab/show/universe">universe</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+of+classes">category of classes</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+with+class+structure">category with class structure</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/constructive+set+theory">constructive set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+set+theory">algebraic set theory</a></li> </ul> </div> <h2 id="foundational_axioms">Foundational axioms</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/foundational+axiom">foundational</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a></strong></p> <ul> <li> <p>basic constructions:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+cartesian+products">axiom of cartesian products</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+disjoint+unions">axiom of disjoint unions</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+the+empty+set">axiom of the empty set</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+fullness">axiom of fullness</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+function+sets">axiom of function sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+quotient+sets">axiom of quotient sets</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/material+set+theory">material axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+anti-foundation">axiom of anti-foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski%27s+axiom">Mostowski's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+transitive+closure">axiom of transitive closure</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+materialization">axiom of materialization</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theoretic axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axioms+of+set+truncation">axioms of set truncation</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/uniqueness+of+identity+proofs">uniqueness of identity proofs</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+K">axiom K</a></li> <li><a class="existingWikiWord" href="/nlab/show/boundary+separation">boundary separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/equality+reflection">equality reflection</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+circle+type+localization">axiom of circle type localization</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theoretic axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+principle">Whitehead's principle</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axioms+of+choice">axioms of choice</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+countable+choice">axiom of countable choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+dependent+choice">axiom of dependent choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+excluded+middle">axiom of excluded middle</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+existence">axiom of existence</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+multiple+choice">axiom of multiple choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/Markov%27s+axiom">Markov's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/presentation+axiom">presentation axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/small+cardinality+selection+axiom">small cardinality selection axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+small+violations+of+choice">axiom of small violations of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+weakly+initial+sets+of+covers">axiom of weakly initial sets of covers</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/large+cardinal+axioms">large cardinal axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+universes">axiom of universes</a></li> <li><a class="existingWikiWord" href="/nlab/show/regular+extension+axiom">regular extension axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/inaccessible+cardinal">inaccessible cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/elementary+embedding">elementary embedding</a></li> <li><a class="existingWikiWord" href="/nlab/show/supercompact+cardinal">supercompact cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/Vop%C4%9Bnka%27s+principle">Vopěnka's principle</a></li> </ul> </li> <li> <p>strong axioms</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+separation">axiom of separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+replacement">axiom of replacement</a></li> </ul> </li> <li> <p>further</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflection+principle">reflection principle</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axiom+of+inequality+spaces">axiom of inequality spaces</a></p> </li> </ul> </div> <h2 id="removing_axioms">Removing axioms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a></li> </ul> <div> <p> <a href="/nlab/edit/foundations+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="vopnkas_principle">Vopěnka's principle</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#statements'>Statements</a></li> <ul> <li><a href='#the_vopnka_principle'>The Vopěnka principle</a></li> <li><a href='#the_weak_vopnka_principle'>The weak Vopěnka principle</a></li> <li><a href='#relativized_versions_of_vopnkas_principle'>Relativized versions of Vopěnka’s principle</a></li> </ul> <li><a href='#motivation'>Motivation</a></li> <li><a href='#Consequences'>Consequences</a></li> <li><a href='#settheoretic_notes'>Set-theoretic notes</a></li> <ul> <li><a href='#first_versus_secondorder'>First- versus second-order</a></li> <li><a href='#vopnka_cardinals'>Vopěnka cardinals</a></li> <li><a href='#definable_counterexamples'>Definable counterexamples</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><strong>Vopěnka’s principle</strong> is a <a class="existingWikiWord" href="/nlab/show/large+cardinal">large cardinal</a> axiom which implies a good deal of simplification in the theory of <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a>.</p> <p>It is fairly strong as large cardinal axioms go: Its consistency follows from the existence of <span class="newWikiWord">huge cardinal<a href="/nlab/new/huge+cardinal">?</a></span>s, and it implies the existence of arbitrarily large <a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a>s.</p> <h2 id="statements">Statements</h2> <h3 id="the_vopnka_principle">The Vopěnka principle</h3> <p>Vopěnka’s principle has many equivalent statements. Here are a few:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>The VP is equivalent to the statement:</p> <p>Every <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete</a> <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> is <a class="existingWikiWord" href="/nlab/show/small+category">small</a>.</p> </div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The VP is equivalent to the statement:</p> <p>For every <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a> sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>M</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mi>α</mi><mo>∈</mo><mi>Ord</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle M_\alpha | \alpha \in Ord\rangle</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/logic">first-order structures</a>, there is a pair of <a class="existingWikiWord" href="/nlab/show/ordinals">ordinals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo><</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha\lt\beta</annotation></semantics></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">M_\alpha</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/elementary+embedding">embeds elementarily</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">M_\beta</annotation></semantics></math>.</p> </div> <div class="num_theorem" id="ColimitsCoreflective"> <h6 id="theorem_3">Theorem</h6> <p>The VP is equivalent to the statement:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a>, every <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>↪</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D \hookrightarrow C</annotation></semantics></math> which is closed under <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s is a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a>.</p> </div> <p>This is (<a href="#AdamekRosicky">AdamekRosicky, theorem 6.28</a>).</p> <div class="num_theorem"> <h6 id="theorem_4">Theorem</h6> <p>The VP is equivalent to the statement:</p> <p>Every <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> (in a slightly more general sense than usual) is a <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a>.</p> </div> <p>This is in (<a href="#Rosicky">Rosicky</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If one insists on the traditional stricter definition of cofibrant generated model category, then the VP still implies that these are all combinatorial. But the VP is slightly stronger than this statement.</p> </div> <div class="num_theorem"> <h6 id="theorem_5">Theorem</h6> <p>The VP is equivalent to both of the statements:</p> <ol> <li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, there exists a <a class="existingWikiWord" href="/nlab/show/C%28n%29-extendible+cardinal">C(n)-extendible cardinal</a>.</li> <li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, there exist arbitrarily large <a class="existingWikiWord" href="/nlab/show/C%28n%29-extendible+cardinals">C(n)-extendible cardinals</a>.</li> </ol> </div> <p>This is in (<a href="#BagariaCasacubertaMathiasRosicky">BCMR</a>).</p> <h3 id="the_weak_vopnka_principle">The weak Vopěnka principle</h3> <p>The Vopěnka principle implies the weak Vopěnka principle.</p> <div class="num_theorem"> <h6 id="theorem_6">Theorem</h6> <p>The weak VP is equivalent to the statement:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a>, every <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>↪</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D \hookrightarrow C</annotation></semantics></math> which is closed under <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s is a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a>.</p> </div> <p>This is <a href="#AdamekRosicky">AdamekRosicky, theorem 6.22 and example 6.23</a></p> <h3 id="relativized_versions_of_vopnkas_principle">Relativized versions of Vopěnka’s principle</h3> <p>Vopěnka’s principle can be relativized to levels of the <a class="existingWikiWord" href="/nlab/show/L%C3%A9vy+hierarchy">Lévy hierarchy</a> by restricting the complexity of the (definable) classes to which it is applied. The following theorems are from (<a href="#BagariaCasacubertaMathiasRosicky">BCMR</a>).</p> <div class="num_theorem"> <h6 id="theorem_7">Theorem</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n\ge 1</annotation></semantics></math>, the following statements are equivalent.</p> <ol> <li>There exists a <a class="existingWikiWord" href="/nlab/show/C%28n%29-extendible+cardinal">C(n)-extendible cardinal</a>.</li> <li>Every proper class of first-order structures that is defined by a conjunction of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\Sigma_{n+1}</annotation></semantics></math> formula and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\Pi_{n+1}</annotation></semantics></math> formula contains distinct structures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/elementary+embedding">elementary embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>↪</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">M\hookrightarrow N</annotation></semantics></math>.</li> </ol> </div> <p>The “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n=0</annotation></semantics></math> case” of this is:</p> <div class="num_theorem"> <h6 id="theorem_8">Theorem</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n\ge 1</annotation></semantics></math>, the following statements are equivalent.</p> <ol> <li>There exists a <a class="existingWikiWord" href="/nlab/show/supercompact+cardinal">supercompact cardinal</a>.</li> <li>Every proper class of first-order structures that is defined by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> formula contains distinct structures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/elementary+embedding">elementary embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>↪</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">M\hookrightarrow N</annotation></semantics></math>.</li> </ol> </div> <p>Many more refined results can be found in (<a href="#BagariaCasacubertaMathiasRosicky">BCMR</a>).</p> <h2 id="motivation">Motivation</h2> <p>From a category-theoretic perspective, Vopěnka’s principle can be motivated by applications and consequences, but it can also be argued for somewhat <em>a priori</em>, on the basis that <em>large discrete categories</em> are rather pathological objects. We can’t avoid them entirely (at least, not without restricting the rest of mathematics fairly severely), but maybe at least we can prevent them from occurring in some nice situations, such as full subcategories of locally presentable categories. See <a href="http://mathoverflow.net/questions/29302/reasons-to-believe-vopenkas-principle-huge-cardinals-are-consistent/29473#29473">this MO answer</a>.</p> <h2 id="Consequences">Consequences</h2> <div class="num_theorem" id="ConsequenceForBousfieldLoc"> <h6 id="theorem_9">Theorem</h6> <p>The VP implies the statement:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/left+proper+model+category">left proper</a> <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z \in Mor(C)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/class">class</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s. Then the <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>Z</mi></msub><mi>W</mi></mrow><annotation encoding="application/x-tex">L_Z W</annotation></semantics></math> exists.</p> </div> <p>This is theorem 2.3 in (<a href="#RosickyTholen">RosickyTholen</a>)</p> <div class="num_corollary" id="ConsequenceForReflectiveInfCatLoc"> <h6 id="corollary">Corollary</h6> <p>The VP implies the statement:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> a class of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Then the reflective <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> extsts.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the facts discussed at <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a> and <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> and <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization of model categories</a> we have that every locally presentable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category is presented by a combinatorial model category and that under this correspondence reflective localizations correspond to left Bousfield localizations. The claim then follows with the (<a href="#ConsequenceForBousfieldLoc">above theorem</a>).</p> </div> <h2 id="settheoretic_notes">Set-theoretic notes</h2> <h3 id="first_versus_secondorder">First- versus second-order</h3> <p>As usually stated, Vopěnka’s principle is not formalizable in first-order <a class="existingWikiWord" href="/nlab/show/ZF">ZF</a> set theory, because it involves a “second-order” <a class="existingWikiWord" href="/nlab/show/quantifier">quantification</a> over <a class="existingWikiWord" href="/nlab/show/proper+classes">proper classes</a> (“…there does not exist a large discrete subcategory…”). It can, however, be formalized in this way in a class-set theory such as <a class="existingWikiWord" href="/nlab/show/NBG">NBG</a>.</p> <p>On the other hand, it can be formalized in ZF as a first-order axiom schema consisting of one axiom for each class-defining formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, stating that “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> does not define a class which is a large discrete subcategory…” We might call this axiom schema the <strong>Vopěnka axiom scheme</strong>. As in most situations of this sort, the first-order Vopěnka scheme is appreciably weaker than the second-order Vopěnka principle. See, for instance, <a href="http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably">this MO question</a> and answer.</p> <h3 id="vopnka_cardinals">Vopěnka cardinals</h3> <p>Unlike some large cardinal axioms, Vopěnka’s principle does not appear to be merely an assertion that “there exist very large cardinals” but rather an assertion about the precise size of the “universe” (the “boundary” between sets and proper classes). In other words, the universe could be “too big” for Vopěnka’s principle to hold, in addition to being “too small.”</p> <p>(The equivalence of Vopěnka’s principle with the existence of <a class="existingWikiWord" href="/nlab/show/C%28n%29-extendible+cardinals">C(n)-extendible cardinals</a> may appear to contradict this. However, the property of being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(n)</annotation></semantics></math>-extendible itself “depends on the size of the whole universe” in a sense.)</p> <p>More precisely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> is a cardinal such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>κ</mi></msub></mrow><annotation encoding="application/x-tex">V_\kappa</annotation></semantics></math> satisfies ZFC + Vopěnka’s principle, then knowing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>></mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\lambda\gt\kappa</annotation></semantics></math> does not necessarily imply that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">V_\lambda</annotation></semantics></math> also satifies Vopěnka’s principle. By contrast, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>κ</mi></msub></mrow><annotation encoding="application/x-tex">V_\kappa</annotation></semantics></math> satisfies ZFC + “there exists a <a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a>” (say), then there must be a measurable cardinal less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>, and that measurable cardinal will still exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">V_\lambda</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>></mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\lambda\gt\kappa</annotation></semantics></math>. On the other hand, large cardinal axioms such as “there exist arbitrarily large measurable cardinals” have the same property that Vopěnka’s principle does: even if measurable cardinals are unbounded below <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>, they will not be unbounded below <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> is the next greatest <a class="existingWikiWord" href="/nlab/show/inaccessible+cardinal">inaccessible cardinal</a> after <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>.</p> <p>Relativizing Vopěnka’s principle to cardinals also raises the same first- versus second-order issues as above. We say that a <strong>Vopěnka cardinal</strong> is one where Vopěnka’s principle holds “in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>κ</mi></msub></mrow><annotation encoding="application/x-tex">V_\kappa</annotation></semantics></math>” where the quantification over classes is interpreted as quantification over all subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>κ</mi></msub></mrow><annotation encoding="application/x-tex">V_\kappa</annotation></semantics></math>. By contrast, we could define an <strong>almost-Vopěnka cardinal</strong> to be one where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>κ</mi></msub></mrow><annotation encoding="application/x-tex">V_\kappa</annotation></semantics></math> satisfies the first-order Vopěnka scheme. Then one can show, using the Mahlo reflection principle (see <a href="http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably/46538#46538">here</a> again), that every Vopěnka cardinal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> is a limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-many almost-Vopěnka cardinals, and in particular the smallest almost-Vopěnka cardinal cannot be Vopěnka. Thus, being Vopěnka is much stronger than being almost-Vopěnka.</p> <h3 id="definable_counterexamples">Definable counterexamples</h3> <p>If Vopěnka’s principle fails, then there exist counterexamples to all of its equivalent statements, such as a large discrete full subcategory of a locally presentable category. If Vopěnka’s principle fails but the first-order Vopěnka scheme holds, then no such counterexamples can be explicitly definable.</p> <p>On the other hand, if the Vopěnka scheme also fails, then there will be explicit finite formulas one can write down which define counterexamples. However, there is no “universal” counterexample, in the following sense: if Vopěnka’s principle is consistent, then for any class-defining formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, there is a model of set theory in which Vopěnka’s principle fails (and even in which the first-order Vopěnka scheme fails), but in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> does not define a counterexample to it. See <a href="http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably/46538#46538">here</a> yet again.</p> <h2 id="references">References</h2> <p>The relation to the theory of <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a> is the contents of chapter 6 of</p> <ul id="AdamekRosicky"> <li><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em><a class="existingWikiWord" href="/nlab/show/Locally+presentable+and+accessible+categories">Locally presentable and accessible categories</a></em>, London Mathematical Society Lecture Note Series 189</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a> is discussed in</p> <ul id="Rosicky"> <li><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em>Are all cofibrantly generated model categories combinatorial?</em> (<a href="http://www.math.muni.cz/~rosicky/papers/cof1.ps">ps</a>)</li> </ul> <p>The implication of VP on <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> and <a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a> are discussed in the following articles</p> <ul id="RosickyTholen"> <li><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <a class="existingWikiWord" href="/nlab/show/Walter+Tholen">Walter Tholen</a>, <em>Left-determined model categories and universal homotopy theories</em> Transactions of the American Mathematical Society <p>Vol. 355, No. 9 (Sep., 2003), pp. 3611-3623 (<a href="http://www.jstor.org/stable/1194855">JSTOR</a>).</p> </li> </ul> <ul id="BagariaCasacubertaMathiasRosicky"> <li> <p><a class="existingWikiWord" href="/nlab/show/Carles+Casacuberta">Carles Casacuberta</a>, Dirk Scevenels, <a class="existingWikiWord" href="/nlab/show/Jeff+Smith">Jeff Smith</a>, <em>Implications of large-cardinal principles in homotopical localization</em> Advances in Mathematics</p> <p>Volume 197, Issue 1, 20 October 2005, Pages 120-139</p> </li> <li> <p>Joan Bagaria, <a class="existingWikiWord" href="/nlab/show/Carles+Casacuberta">Carles Casacuberta</a>, Adrian Mathias, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosicky">Jiří Rosicky</a> <em>Definable orthogonality classes in accessible categories are small</em>, <a href="http://arxiv.org/abs/1101.2792">arXiv</a></p> </li> </ul> <ul> <li>Giulio Lo Monaco, <em>Vopěnka’s principle in ∞-categories</em>, <a href="https://arxiv.org/abs/2105.04251">arxiv:2105.04251</a></li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/large+cardinal">large cardinal</a> strength of the weak Vopěnka principle is discussed in</p> <ul> <li><span class="newWikiWord">Trevor M. Wilson<a href="/nlab/new/Trevor+M.+Wilson">?</a></span>, <em>The large cardinal strength of Weak Vopěnka’s Principle</em>, <a href="https://arxiv.org/abs/1907.00284">arXiv</a>.</li> </ul> <p>The following paper shows that weak Vopěnka’s principle is indeed weaker than Vopěnka’s principle:</p> <ul> <li><span class="newWikiWord">Trevor M. Wilson<a href="/nlab/new/Trevor+M.+Wilson">?</a></span>, <em>Weak Vopěnka’s Principle does not imply Vopěnka’s Principle</em>, <a href="https://arxiv.org/abs/1909.09333">arXiv</a>.</li> </ul> <p>Applications to localizations of presentable (∞,1)-categories are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Carles+Casacuberta">Carles Casacuberta</a>, <a class="existingWikiWord" href="/nlab/show/Javier+J.+Guti%C3%A9rrez">Javier J. Gutiérrez</a>, <em>Homotopy reflectivity is equivalent to the weak Vopěnka principle</em>, <a href="https://arxiv.org/abs/2410.21244">arXiv</a>.</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/foundational+axiom">foundational axiom</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on November 5, 2024 at 00:27:34. 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