CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> theorem in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> theorem </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Theorems</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="foundations">Foundations</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/foundations">foundations</a></strong></p> <h2 id="the_basis_of_it_all">The basis of it all</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mathematical+logic">mathematical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deduction+system">deduction system</a>, <a 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> <h4 id="type_theory">Type theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></strong> <a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a>, <a class="existingWikiWord" href="/nlab/show/practical+foundations">practical foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a>, <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/antecedents">antecedents</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/consequent">consequent</a>, <a class="existingWikiWord" href="/nlab/show/succedents">succedents</a></li> </ul> </li> </ul> <ol> <li><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+introduction+rule">term introduction rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/computation+rule">computation rule</a></li> </ol> <p><strong><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent</a>, <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></strong> <a class="existingWikiWord" href="/nlab/show/object+language">object language</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a>, <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a>/<a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></strong> = <br /> <strong><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/programs+as+proofs">programs as proofs</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation type theory/category theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/logic">logic</a></th><th><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> (<a class="existingWikiWord" href="/nlab/show/internal+logic+of+set+theory">internal logic</a> of)</th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/predicate">predicate</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/family+of+sets">family of sets</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/display+morphism">display morphism</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof">proof</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/element">element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term">term</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+rule">cut rule</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/classifying+morphisms">classifying morphisms</a> / <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/display+maps">display maps</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/substitution">substitution</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/introduction+rule">introduction rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/counit">counit</a> for hom-tensor adjunction</td><td style="text-align: left;">lambda</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elimination+rule">elimination rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> for hom-tensor adjunction</td><td style="text-align: left;">application</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+elimination">cut elimination</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">one of the <a class="existingWikiWord" href="/nlab/show/zigzag+identities">zigzag identities</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/beta+reduction">beta reduction</a></td></tr> <tr><td style="text-align: left;">identity elimination for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">the other <a class="existingWikiWord" href="/nlab/show/zigzag+identity">zigzag identity</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/eta+conversion">eta conversion</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/singleton">singleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-2%29-truncated+object">(-2)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+0">h-level 0</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/false">false</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>, <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncated+object">(-1)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>, <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product">product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product+type">product type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> (into <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (into <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (into <a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> into <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> into <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> into <a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subsingletons">subsingletons</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (of family of <a class="existingWikiWord" href="/nlab/show/h-propositions">h-propositions</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bijection+set">bijection set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+of+isomorphisms">object of isomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+type">equivalence type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+set">support set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+object">support object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/propositional+truncation">propositional truncation</a>/<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-image">n-image</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> into <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/n-truncation">n-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/diagonal+function">diagonal function</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+subset">diagonal subset</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+relation">diagonal relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>/<a class="existingWikiWord" href="/nlab/show/path+type">path type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completely+presented+set">completely presented set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>/<a class="existingWikiWord" href="/nlab/show/0-truncated+object">0-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+2">h-level 2</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a>/<a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> with <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">internal 0-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>/<a class="existingWikiWord" href="/nlab/show/setoid">setoid</a> with its <a class="existingWikiWord" href="/nlab/show/pseudo-equivalence+relation">pseudo-equivalence relation</a> an actual <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a>/<a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+type">quotient type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/W-type">W-type</a>, <a class="existingWikiWord" href="/nlab/show/M-type">M-type</a></td></tr> <tr><td style="text-align: left;">higher <a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></td></tr> <tr><td style="text-align: left;">-</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+inductive+type">quotient inductive type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/limit">limit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinductive+type">coinductive type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/preset">preset</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a> without <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+of+propositions">type of propositions</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+of+discourse">domain of discourse</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universe">universe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modality">modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a>, (<a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a>) <a class="existingWikiWord" href="/nlab/show/monad">monad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a>, <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof+net">proof net</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></td></tr> <tr><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/contraction+rule">contraction rule</a></td><td style="text-align: left;"></td><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/synthetic+mathematics">synthetic mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+specific+embedded+programming+language">domain specific embedded programming language</a></td></tr> </tbody></table> </div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+levels">homotopy levels</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/michaelshulman/show/2-categorical+logic">2-categorical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory+-+contents">homotopy type theory - contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a>, <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/display+map">display map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+a+topos">internal logic of a topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell-Benabou+language">Mitchell-Benabou language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type-theoretic+model+category">type-theoretic model category</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/type+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <blockquote> <p>There are good reasons why the theorems should all be easy and the definitions hard. (<a class="existingWikiWord" href="/nlab/show/Michael+Spivak">Michael Spivak</a>, preface to “<a class="existingWikiWord" href="/nlab/show/Calculus+on+Manifolds">Calculus on Manifolds</a>” )</p> </blockquote> <h1 id="theorems">Theorems</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#quotations'>Quotations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In the traditional language of <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>, a <em>theorem</em> is a statement which is of interest in its own right and which has been proven to be true, though the proof may not be immediately obvious. This contrasts with a <em><a class="existingWikiWord" href="/nlab/show/lemma">lemma</a></em> (which is usually of interest primarily because of its implications for other statements), a <em><a class="existingWikiWord" href="/nlab/show/conjecture">conjecture</a></em> (which has not yet been proved), an <em><a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></em> (which is obviously true or assumed to be true), a <em>definition</em> (which becomes true by virtue of its assigning meaning to a word or phrase), a <em><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a></em> (which usually follows more easily from known facts than a theorem does), or a <em>corollary</em> (which follows immediately from facts recently proven).</p> <p>The discipline of <a class="existingWikiWord" href="/nlab/show/logic">logic</a> formalizes the notion of <a class="existingWikiWord" href="/nlab/show/proof">proof</a>, but not the notions of “interest” or “immediacy”. Thus, to a logician, <em>any</em> proved statement is often called a theorem. (Mathematicians know this meaning too, but still usually reserve the term ‘theorem’ for important theorems in their published work.) The term ‘<a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>’, to a logician, means <em>any</em> statement and does <em>not</em> imply the existence of a proof. The term ‘<a class="existingWikiWord" href="/nlab/show/axiom">axiom</a>’ is used in a way that somewhat matches its ordinary usage, but as a logician counts trivial proofs as proofs, an axiom is also a special case of a theorem. Logic rarely studies <a class="existingWikiWord" href="/nlab/show/definition">definition</a>s explicitly, but in some theories they do play a role, similar to their informal usage. The other terms appear not to be used in logic.</p> <h2 id="definition">Definition</h2> <p>In a given <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, in a given <a class="existingWikiWord" href="/nlab/show/context">context</a>, we have various <a class="existingWikiWord" href="/nlab/show/propositions">propositions</a> and various <a class="existingWikiWord" href="/nlab/show/proofs">proofs</a> of propositions. In that context, a <strong>theorem</strong> is a proposition with a proof.</p> <p>Classically, a theorem is a proposition for which there <em>exists</em> a proof, but in some contexts (such as, perhaps, fully formalized <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>), one may use “theorem” to mean a proposition <em>together with</em> a proof.</p> <p>A theorem should be contrasted with a <a class="existingWikiWord" href="/nlab/show/tautology">tautology</a>: a proposition that is <a class="existingWikiWord" href="/nlab/show/truth">true</a> in all <a class="existingWikiWord" href="/nlab/show/models">models</a>. If every theorem in a given logic is a tautology in a given class of models for that logic, then we say that the class of models is <strong>sound</strong> for that logic; if conversely every tautology is a theorem, then we say that the class of models is <strong>complete</strong>.</p> <h2 id="examples">Examples</h2> <p>… we might list famous important theorems/lemmas/etc in the <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>Lab here …</p> <h2 id="quotations">Quotations</h2> <ul> <li> <p>A mathematician is a device for turning coffee into theorems. —Alfréd Rényi</p> </li> <li> <p>Lemmas do the work in mathematics: Theorems, like management, just take the credit. —<a class="existingWikiWord" href="/nlab/show/Paul+Taylor">Paul Taylor</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>mathematical statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a>, <a class="existingWikiWord" href="/nlab/show/assertion">assertion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothesis">hypothesis</a>, <a class="existingWikiWord" href="/nlab/show/consequence">consequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/assignment">assignment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a> (<a class="existingWikiWord" href="/nlab/show/inductive+definition">inductive</a>, <a class="existingWikiWord" href="/nlab/show/coinductive+definition">coinductive</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lemma">lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/example">example</a>, <a class="existingWikiWord" href="/nlab/show/counterexample">counterexample</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conjecture">conjecture</a>, <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exercise">exercise</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analogy">analogy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautology">tautology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paradox">paradox</a></p> </li> </ul> </div> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Hales">Thomas Hales</a>, <em>Formal proof</em> (<a href="http://www.ams.org/notices/200811/tx081101370p.pdf">pdf</a>)</p> </li> <li> <p>John Harrison, <em>Formal proof – theory and practice</em> (<a href="http://www.ams.org/notices/200811/tx081101395p.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 5, 2023 at 13:32:11. See the <a href="/nlab/history/theorem" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/theorem" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/theorem/18" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/theorem" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/theorem" accesskey="S" class="navlink" id="history" rel="nofollow">History (18 revisions)</a> <a href="/nlab/show/theorem/cite" style="color: black">Cite</a> <a href="/nlab/print/theorem" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/theorem" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>