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vector-toc-level-2"> <a class="vector-toc-link" href="#Consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Consistency</span> </div> </a> <ul id="toc-Consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Utility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Utility"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Utility</span> </div> </a> <ul id="toc-Utility-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sets,_membership_and_equality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sets,_membership_and_equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Sets, membership and equality</span> </div> </a> <button aria-controls="toc-Sets,_membership_and_equality-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Sets, membership and equality subsection</span> </button> <ul id="toc-Sets,_membership_and_equality-sublist" class="vector-toc-list"> <li id="toc-Note_on_consistency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Note_on_consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Note on consistency</span> </div> </a> <ul id="toc-Note_on_consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Membership" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Membership"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Membership</span> </div> </a> <ul id="toc-Membership-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Equality</span> </div> </a> <ul id="toc-Equality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Empty_set" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Empty_set"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Empty set</span> </div> </a> <ul id="toc-Empty_set-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Specifying_sets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Specifying_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Specifying sets</span> </div> </a> <ul id="toc-Specifying_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subsets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Subsets"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Subsets</span> </div> </a> <ul id="toc-Subsets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Universal_sets_and_absolute_complements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Universal_sets_and_absolute_complements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Universal sets and absolute complements</span> </div> </a> <ul id="toc-Universal_sets_and_absolute_complements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unions,_intersections,_and_relative_complements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unions,_intersections,_and_relative_complements"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Unions, intersections, and relative complements</span> </div> </a> <ul id="toc-Unions,_intersections,_and_relative_complements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordered_pairs_and_Cartesian_products" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ordered_pairs_and_Cartesian_products"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Ordered pairs and Cartesian products</span> </div> </a> <ul id="toc-Ordered_pairs_and_Cartesian_products-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Some_important_sets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Some_important_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Some important sets</span> </div> </a> <ul id="toc-Some_important_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paradoxes_in_early_set_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Paradoxes_in_early_set_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Paradoxes in early set theory</span> </div> </a> <ul id="toc-Paradoxes_in_early_set_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 31 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-31" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">31 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%85%D8%AC%D9%85%D9%88%D8%B9%D8%A7%D8%AA_%D8%A7%D9%84%D9%85%D8%A8%D8%B3%D8%B7%D8%A9" title="نظرية المجموعات المبسطة – Arabic" lang="ar" hreflang="ar" data-title="نظرية المجموعات المبسطة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_informal_de_conxuntos" title="Teoría informal de conxuntos – Asturian" lang="ast" hreflang="ast" data-title="Teoría informal de conxuntos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_informal_de_conjunts" title="Teoria informal de conjunts – Catalan" lang="ca" hreflang="ca" data-title="Teoria informal de conjunts" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Naivn%C3%AD_teorie_mno%C5%BEin" title="Naivní teorie množin – Czech" lang="cs" hreflang="cs" data-title="Naivní teorie množin" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Naive_Mengenlehre" title="Naive Mengenlehre – German" lang="de" hreflang="de" data-title="Naive Mengenlehre" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Naiivne_hulgateooria" title="Naiivne hulgateooria – Estonian" lang="et" hreflang="et" data-title="Naiivne hulgateooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%86%CE%B5%CE%BB%CE%AE%CF%82_%CF%83%CF%85%CE%BD%CE%BF%CE%BB%CE%BF%CE%B8%CE%B5%CF%89%CF%81%CE%AF%CE%B1" title="Αφελής συνολοθεωρία – Greek" lang="el" hreflang="el" data-title="Αφελής συνολοθεωρία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_informal_de_conjuntos" title="Teoría informal de conjuntos – Spanish" lang="es" hreflang="es" data-title="Teoría informal de conjuntos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Naiva_aroteorio" title="Naiva aroteorio – Esperanto" lang="eo" hreflang="eo" data-title="Naiva aroteorio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Multzoen_teoria_informal" title="Multzoen teoria informal – Basque" lang="eu" hreflang="eu" data-title="Multzoen teoria informal" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C_%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87%E2%80%8C%D9%87%D8%A7" title="نظریه طبیعی مجموعه‌ها – Persian" lang="fa" hreflang="fa" data-title="نظریه طبیعی مجموعه‌ها" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_na%C3%AFve_des_ensembles" title="Théorie naïve des ensembles – French" lang="fr" hreflang="fr" data-title="Théorie naïve des ensembles" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_informal_de_conxuntos" title="Teoría informal de conxuntos – Galician" lang="gl" hreflang="gl" data-title="Teoría informal de conxuntos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%86%8C%EB%B0%95%ED%95%9C_%EC%A7%91%ED%95%A9%EB%A1%A0" title="소박한 집합론 – Korean" lang="ko" hreflang="ko" data-title="소박한 집합론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Naivna_teorija_skupova" title="Naivna teorija skupova – Croatian" lang="hr" hreflang="hr" data-title="Naivna teorija skupova" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_ingenua_degli_insiemi" title="Teoria ingenua degli insiemi – Italian" lang="it" hreflang="it" data-title="Teoria ingenua degli insiemi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%91%D7%95%D7%A6%D7%95%D7%AA_%D7%94%D7%A0%D7%90%D7%99%D7%91%D7%99%D7%AA" title="תורת הקבוצות הנאיבית – Hebrew" lang="he" hreflang="he" data-title="תורת הקבוצות הנאיבית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Naiv_halmazelm%C3%A9let" title="Naiv halmazelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Naiv halmazelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B8%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D1%82%D0%B0" title="Наивна теорија на множествата – Macedonian" lang="mk" hreflang="mk" data-title="Наивна теорија на множествата" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9B%E1%80%AD%E1%80%AF%E1%80%B8%E1%80%9B%E1%80%AD%E1%80%AF%E1%80%B8%E1%80%A1%E1%80%85%E1%80%AF%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE" title="ရိုးရိုးအစုသီအိုရီ – Burmese" lang="my" hreflang="my" data-title="ရိုးရိုးအစုသီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Na%C3%AFeve_verzamelingenleer" title="Naïeve verzamelingenleer – Dutch" lang="nl" hreflang="nl" data-title="Naïeve verzamelingenleer" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B4%A0%E6%9C%B4%E9%9B%86%E5%90%88%E8%AB%96" title="素朴集合論 – Japanese" lang="ja" hreflang="ja" data-title="素朴集合論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_ing%C3%AAnua_dos_conjuntos" title="Teoria ingênua dos conjuntos – Portuguese" lang="pt" hreflang="pt" data-title="Teoria ingênua dos conjuntos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B8%D0%B2%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2" title="Наивная теория множеств – Russian" lang="ru" hreflang="ru" data-title="Наивная теория множеств" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Naive_set_theory" title="Naive set theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Naive set theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Naivna_teorija_skupova" title="Naivna teorija skupova – Serbian" lang="sr" hreflang="sr" data-title="Naivna teorija skupova" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a 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.hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical topic. For the book of the same name, see <a href="/wiki/Naive_Set_Theory_(book)" title="Naive Set Theory (book)">Naive Set Theory (book)</a>.</div> <p><b>Naive set theory</b> is any of several theories of sets used in the discussion of the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Unlike <a href="/wiki/Set_theory#Axiomatic_set_theory" title="Set theory">axiomatic set theories</a>, which are defined using <a href="/wiki/Mathematical_logic#Formal_logical_systems" title="Mathematical logic">formal logic</a>, naive set theory is defined informally, in <a href="/wiki/Natural_language" title="Natural language">natural language</a>. It describes the aspects of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">mathematical sets</a> familiar in <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a> (for example <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagrams</a> and symbolic reasoning about their <a href="/wiki/Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a>), and suffices for the everyday use of set theory concepts in contemporary mathematics.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (<a href="/wiki/Number" title="Number">numbers</a>, <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a>, <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping stone towards more formal treatments. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Method">Method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=1" title="Edit section: Method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>naive theory</i> in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses <a href="/wiki/Natural_language" title="Natural language">natural language</a> to describe sets and operations on sets. Such theory treats sets as platonic absolute objects. The words <i>and</i>, <i>or</i>, <i>if ... then</i>, <i>not</i>, <i>for some</i>, <i>for every</i> are treated as in ordinary mathematics. As a matter of convenience, use of naive set theory and its formalism prevails even in higher mathematics – including in more formal settings of set theory itself. </p><p>The first development of <a href="/wiki/Set_theory" title="Set theory">set theory</a> was a naive set theory. It was created at the end of the 19th century by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> as part of his study of <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a><sup id="cite_ref-FOOTNOTECantor1874_5-0" class="reference"><a href="#cite_note-FOOTNOTECantor1874-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and developed by <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a> in his <i>Grundgesetze der Arithmetik</i>. </p><p>Naive set theory may refer to several very distinct notions. It may refer to </p> <ul><li>Informal presentation of an axiomatic set theory, e.g. as in <i><a href="/wiki/Naive_Set_Theory_(book)" title="Naive Set Theory (book)">Naive Set Theory</a></i> by <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>.</li> <li>Early or later versions of <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>'s theory and other informal systems.</li> <li>Decidedly inconsistent theories (whether axiomatic or not), such as a theory of <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> that yielded <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>, and theories of <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Paradoxes">Paradoxes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=2" title="Edit section: Paradoxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The assumption that any property may be used to form a set, without restriction, leads to <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a>. One common example is <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>: there is no set consisting of "all sets that do not contain themselves". Thus consistent systems of naive set theory must include some limitations on the principles which can be used to form sets. </p> <div class="mw-heading mw-heading3"><h3 id="Cantor's_theory"><span id="Cantor.27s_theory"></span>Cantor's theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=3" title="Edit section: Cantor's theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some believe that <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>'s set theory was not actually implicated in the set-theoretic paradoxes (see Frápolli 1991). One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">Cantor's paradox</a><sup id="cite_ref-Letter_to_Hilbert_8-0" class="reference"><a href="#cite_note-Letter_to_Hilbert-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and did not believe that they discredited his theory.<sup id="cite_ref-Letters_to_Dedekind_10-0" class="reference"><a href="#cite_note-Letters_to_Dedekind-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Cantor's paradox can actually be derived from the above (false) assumption—that any property <span class="texhtml"><i>P</i>(<i>x</i>)</span> may be used to form a set—using for <span class="texhtml"><i>P</i>(<i>x</i>)</span> "<span class="texhtml mvar" style="font-style:italic;">x</span> is a <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a>". Frege explicitly axiomatized a theory in which a formalized version of naive set theory can be interpreted, and it is <i>this</i> formal theory which <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> actually addressed when he presented his paradox, not necessarily a theory Cantor—who, as mentioned, was aware of several paradoxes—presumably had in mind. </p> <div class="mw-heading mw-heading3"><h3 id="Axiomatic_theories">Axiomatic theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=4" title="Edit section: Axiomatic theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when. </p> <div class="mw-heading mw-heading3"><h3 id="Consistency">Consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=5" title="Edit section: Consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A naive set theory is not <i>necessarily</i> inconsistent, if it correctly specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms. It is possible to state all the axioms explicitly, as in the case of Halmos' <i>Naive Set Theory</i>, which is actually an informal presentation of the usual axiomatic <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>. It is "naive" in that the language and notations are those of ordinary informal mathematics, and in that it does not deal with consistency or completeness of the axiom system. </p><p>Likewise, an axiomatic set theory is not necessarily consistent: not necessarily free of paradoxes. It follows from <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> that a sufficiently complicated <a href="/wiki/First_order_logic" class="mw-redirect" title="First order logic">first order logic</a> system (which includes most common axiomatic set theories) cannot be proved consistent from within the theory itself – even if it actually is consistent. However, the common axiomatic systems are generally believed to be consistent; by their axioms they do exclude <i>some</i> paradoxes, like <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>. Based on <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's theorem</a>, it is just not known – and never can be – if there are <i>no</i> paradoxes at all in these theories or in any first-order set theory. </p><p>The term <i>naive set theory</i> is still today also used in some literature<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory. </p> <div class="mw-heading mw-heading3"><h3 id="Utility">Utility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=6" title="Edit section: Utility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The choice between an axiomatic approach and other approaches is largely a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory. References to particular axioms typically then occur only when demanded by tradition, e.g. the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> is often mentioned when used. Likewise, formal proofs occur only when warranted by exceptional circumstances. This informal usage of axiomatic set theory can have (depending on notation) precisely the <i>appearance</i> of naive set theory as outlined below. It is considerably easier to read and write (in the formulation of most statements, proofs, and lines of discussion) and is less error-prone than a strictly formal approach. </p> <div class="mw-heading mw-heading2"><h2 id="Sets,_membership_and_equality"><span id="Sets.2C_membership_and_equality"></span>Sets, membership and equality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=7" title="Edit section: Sets, membership and equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In naive set theory, a <b>set</b> is described as a well-defined collection of objects. These objects are called the <b>elements</b> or <b>members</b> of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even <a href="/wiki/Integer" title="Integer">integers</a>. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Passage_with_the_set_definition_of_Georg_Cantor.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/250px-Passage_with_the_set_definition_of_Georg_Cantor.png" decoding="async" width="250" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/375px-Passage_with_the_set_definition_of_Georg_Cantor.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/500px-Passage_with_the_set_definition_of_Georg_Cantor.png 2x" data-file-width="1401" data-file-height="594" /></a><figcaption>Passage with the original set definition of Georg Cantor</figcaption></figure> <p>The definition of sets goes back to <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>. He wrote in his 1915 article <i><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141020034245/http://gdz.sub.uni-goettingen.de/index.php?id=pdf&no_cache=1&IDDOC=36218">Beiträge zur Begründung der transfiniten Mengenlehre</a></i>: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.</p><div class="templatequotecite">— <cite>Georg Cantor</cite></div></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.</p><div class="templatequotecite">— <cite>Georg Cantor</cite></div></blockquote> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:First_usage_of_the_symbol_%E2%88%88.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/First_usage_of_the_symbol_%E2%88%88.png/250px-First_usage_of_the_symbol_%E2%88%88.png" decoding="async" width="250" height="74" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/First_usage_of_the_symbol_%E2%88%88.png/375px-First_usage_of_the_symbol_%E2%88%88.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/First_usage_of_the_symbol_%E2%88%88.png/500px-First_usage_of_the_symbol_%E2%88%88.png 2x" data-file-width="2508" data-file-height="740" /></a><figcaption>First usage of the symbol ϵ in the work <i><a rel="nofollow" class="external text" href="https://archive.org/details/arithmeticespri00peangoog">Arithmetices principia nova methodo exposita</a></i> by <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Note_on_consistency">Note on consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=8" title="Edit section: Note on consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It does <i>not</i> follow from this definition <i>how</i> sets can be formed, and what operations on sets again will produce a set. The term "well-defined" in "well-defined collection of objects" cannot, by itself, guarantee the consistency and unambiguity of what exactly constitutes and what does not constitute a set. Attempting to achieve this would be the realm of axiomatic set theory or of axiomatic <b>class theory</b>. </p><p>The problem, in this context, with informally formulated set theories, not derived from (and implying) any particular axiomatic theory, is that there may be several widely differing formalized versions, that have both different sets and different rules for how new sets may be formed, that all conform to the original informal definition. For example, Cantor's verbatim definition allows for considerable freedom in what constitutes a set. On the other hand, it is unlikely that Cantor was particularly interested in sets containing cats and dogs, but rather only in sets containing purely mathematical objects. An example of such a class of sets could be the <a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">von Neumann universe</a>. But even when fixing the class of sets under consideration, it is not always clear which rules for set formation are allowed without introducing paradoxes. </p><p>For the purpose of fixing the discussion below, the term "well-defined" should instead be interpreted as an <i>intention</i>, with either implicit or explicit rules (axioms or definitions), to rule out inconsistencies. The purpose is to keep the often deep and difficult issues of consistency away from the, usually simpler, context at hand. An explicit ruling out of <i>all</i> conceivable inconsistencies (paradoxes) cannot be achieved for an axiomatic set theory anyway, due to Gödel's second incompleteness theorem, so this does not at all hamper the utility of naive set theory as compared to axiomatic set theory in the simple contexts considered below. It merely simplifies the discussion. Consistency is henceforth taken for granted unless explicitly mentioned. </p> <div class="mw-heading mw-heading3"><h3 id="Membership">Membership</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=9" title="Edit section: Membership"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>x</i> is a member of a set <i>A</i>, then it is also said that <i>x</i> <b>belongs to</b> <i>A</i>, or that <i>x</i> is in <i>A</i>. This is denoted by <i>x</i> ∈ <i>A</i>. The symbol ∈ is a derivation from the lowercase Greek letter <a href="/wiki/Epsilon" title="Epsilon">epsilon</a>, "ε", introduced by <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> in 1889 and is the first letter of the word <a href="https://en.wiktionary.org/wiki/%E1%BC%90%CF%83%CF%84%CE%AF" class="extiw" title="wikt:ἐστί">ἐστί</a> (means "is"). The symbol ∉ is often used to write <i>x</i> ∉ <i>A</i>, meaning "x is not in A". </p> <div class="mw-heading mw-heading3"><h3 id="Equality">Equality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=10" title="Edit section: Equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two sets <i>A</i> and <i>B</i> are defined to be <b><a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a></b> when they have precisely the same elements, that is, if every element of <i>A</i> is an element of <i>B</i> and every element of <i>B</i> is an element of <i>A</i>. (See <a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">axiom of extensionality</a>.) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> less than 6. If the sets <i>A</i> and <i>B</i> are equal, this is denoted symbolically as <i>A</i> = <i>B</i> (as usual). </p> <div class="mw-heading mw-heading3"><h3 id="Empty_set">Empty set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=11" title="Edit section: Empty set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Empty_set" title="Empty set">empty set</a>, denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span> and sometimes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6f1caa524dfcc90158ad69a51b5f9577fe5f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \{\}}"></span>, is a set with no members at all. Because a set is determined completely by its elements, there can be only one empty set. (See <a href="/wiki/Axiom_of_empty_set" title="Axiom of empty set">axiom of empty set</a>.)<sup id="cite_ref-FOOTNOTEHalmos19749_12-0" class="reference"><a href="#cite_note-FOOTNOTEHalmos19749-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Although the empty set has no members, it can be a member of other sets. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing \neq \{\varnothing \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo>≠</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing \neq \{\varnothing \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5898eaad76efe9f3859324797a158e0fc79b53b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.04ex; height:2.843ex;" alt="{\displaystyle \varnothing \neq \{\varnothing \}}"></span>, because the former has no members and the latter has one member.<sup id="cite_ref-FOOTNOTEHalmos197410_13-0" class="reference"><a href="#cite_note-FOOTNOTEHalmos197410-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Specifying_sets">Specifying sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=12" title="Edit section: Specifying sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest way to describe a set is to list its elements between curly braces (known as defining a set <i>extensionally</i>). Thus <span class="texhtml">{1, 2}</span> denotes the set whose only elements are <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1</span> and <span class="nowrap"><span data-sort-value="7000200000000000000♠"></span>2</span>. (See <a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">axiom of pairing</a>.) Note the following points: </p> <ul><li>The order of elements is immaterial; for example, <span class="texhtml">{1, 2} = {2, 1}</span>.</li> <li>Repetition (<a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicity</a>) of elements is irrelevant; for example, <span class="texhtml">{1, 2, 2} = {1, 1, 1, 2} = {1, 2}</span>.</li></ul> <p>(These are consequences of the definition of equality in the previous section.) </p><p>This notation can be informally abused by saying something like <span class="texhtml">{dogs}</span> to indicate the set of all dogs, but this example would usually be read by mathematicians as "the set containing the single element <i>dogs</i>". </p><p>An extreme (but correct) example of this notation is <span class="texhtml">{}</span>, which denotes the empty set. </p><p>The notation <span class="texhtml">{<i>x</i> : <i>P</i>(<i>x</i>)}</span>, or sometimes <span class="texhtml">{<i>x</i> |<i>P</i>(<i>x</i>)}</span>, is used to denote the set containing all objects for which the condition <span class="texhtml mvar" style="font-style:italic;">P</span> holds (known as defining a set <i>intensionally</i>). For example, <span class="texhtml">{<i>x</i> | <i>x</i> ∈ <b>R</b>}</span> denotes the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>, <span class="texhtml">{<i>x</i> | <i>x</i> has blonde hair}</span> denotes the set of everything with blonde hair. </p><p>This notation is called <a href="/wiki/Set-builder_notation" title="Set-builder notation">set-builder notation</a> (or "<b>set comprehension</b>", particularly in the context of <a href="/wiki/Functional_programming" title="Functional programming">Functional programming</a>). Some variants of set builder notation are: </p> <ul><li><span class="texhtml">{<i>x</i> ∈ <i>A</i> | <i>P</i>(<i>x</i>)}</span> denotes the set of all <span class="texhtml mvar" style="font-style:italic;">x</span> that are already members of <span class="texhtml mvar" style="font-style:italic;">A</span> such that the condition <span class="texhtml mvar" style="font-style:italic;">P</span> holds for <span class="texhtml mvar" style="font-style:italic;">x</span>. For example, if <span class="texhtml"><b>Z</b></span> is the set of <a href="/wiki/Integer" title="Integer">integers</a>, then <span class="texhtml">{<i>x</i> ∈ <b>Z</b> | <i>x</i> is even}</span> is the set of all <a href="/wiki/Even_and_odd_numbers" class="mw-redirect" title="Even and odd numbers">even</a> integers. (See <a href="/wiki/Axiom_of_specification" class="mw-redirect" title="Axiom of specification">axiom of specification</a>.)</li> <li><span class="texhtml">{<i>F</i>(<i>x</i>) | <i>x</i> ∈ <i>A</i>}</span> denotes the set of all objects obtained by putting members of the set <span class="texhtml mvar" style="font-style:italic;">A</span> into the formula <span class="texhtml mvar" style="font-style:italic;">F</span>. For example, <span class="texhtml">{2<i>x</i> | <i>x</i> ∈ <b>Z</b>}</span> is again the set of all even integers. (See <a href="/wiki/Axiom_of_replacement" class="mw-redirect" title="Axiom of replacement">axiom of replacement</a>.)</li> <li><span class="texhtml">{<i>F</i>(<i>x</i>) | <i>P</i>(<i>x</i>)}</span> is the most general form of set builder notation. For example, <span class="texhtml">{<i>x'</i>s owner | <i>x</i> is a dog}</span> is the set of all dog owners.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Subsets">Subsets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=13" title="Edit section: Subsets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given two sets <i>A</i> and <i>B</i>, <i>A</i> is a <b><a href="/wiki/Subset" title="Subset">subset</a></b> of <i>B</i> if every element of <i>A</i> is also an element of <i>B</i>. In particular, each set <i>B</i> is a subset of itself; a subset of <i>B</i> that is not equal to <i>B</i> is called a <b>proper subset</b>. </p><p>If <i>A</i> is a subset of <i>B</i>, then one can also say that <i>B</i> is a <b>superset</b> of <i>A</i>, that <i>A</i> is <b>contained in</b> <i>B</i>, or that <i>B</i> <b>contains</b> <i>A</i>. In symbols, <span class="texhtml"><i>A</i> ⊆ <i>B</i></span> means that <i>A</i> is a subset of <i>B</i>, and <span class="texhtml"><i>B</i> ⊇ <i>A</i></span> means that <i>B</i> is a superset of <i>A</i>. Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for <i>proper</i> subsets. For clarity, one can explicitly use the symbols ⊊ and ⊋ to indicate non-equality. </p><p>As an illustration, let <b>R</b> be the set of real numbers, let <b>Z</b> be the set of integers, let <i>O</i> be the set of odd integers, and let <i>P</i> be the set of current or former <a href="/wiki/President_of_the_United_States" title="President of the United States">U.S. Presidents</a>. Then <i>O</i> is a subset of <b>Z</b>, <b>Z</b> is a subset of <b>R</b>, and (hence) <i>O</i> is a subset of <b>R</b>, where in all cases <i>subset</i> may even be read as <i>proper subset</i>. Not all sets are comparable in this way. For example, it is not the case either that <b>R</b> is a subset of <i>P</i> nor that <i>P</i> is a subset of <b>R</b>. </p><p>It follows immediately from the definition of equality of sets above that, given two sets <i>A</i> and <i>B</i>, <span class="texhtml"><i>A</i> = <i>B</i></span> if and only if <span class="texhtml"><i>A</i> ⊆ <i>B</i></span> and <span class="texhtml"><i>B</i> ⊆ <i>A</i></span>. In fact this is often given as the definition of equality. Usually when trying to <a href="/wiki/Mathematical_proof" title="Mathematical proof">prove</a> that two sets are equal, one aims to show these two inclusions. The <a href="/wiki/Empty_set" title="Empty set">empty set</a> is a subset of every set (the statement that all elements of the empty set are also members of any set <i>A</i> is <a href="/wiki/Vacuously_true" class="mw-redirect" title="Vacuously true">vacuously true</a>). </p><p>The set of all subsets of a given set <i>A</i> is called the <b><a href="/wiki/Power_set" title="Power set">power set</a></b> of <i>A</i> and is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6d28a1b787f8c321de35ccc9305fd6cbda9934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.676ex;" alt="{\displaystyle 2^{A}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f264d19e21604793c6dc54f8044df454db82744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}"></span>; the "<span class="texhtml mvar" style="font-style:italic;">P</span>" is sometimes in a <a href="/wiki/Script_(typefaces)" class="mw-redirect" title="Script (typefaces)">script</a> font: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32728013f461acdf9fc95ff820f251a56c854129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.031ex; height:2.843ex;" alt="{\displaystyle \wp (A)}"></span>⁠</span>. If the set <i>A</i> has <i>n</i> elements, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f264d19e21604793c6dc54f8044df454db82744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}"></span> will have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"></span> elements. </p> <div class="mw-heading mw-heading2"><h2 id="Universal_sets_and_absolute_complements">Universal sets and absolute complements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=14" title="Edit section: Universal sets and absolute complements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In certain contexts, one may consider all sets under consideration as being subsets of some given <a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">universal set</a>. For instance, when investigating properties of the <a href="/wiki/Real_number" title="Real number">real numbers</a> <b>R</b> (and subsets of <b>R</b>), <b>R</b> may be taken as the universal set. A true universal set is not included in standard set theory (see <b><a href="#Paradoxes">Paradoxes</a></b> below), but is included in some non-standard set theories. </p><p>Given a universal set <b>U</b> and a subset <i>A</i> of <b>U</b>, the <b><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></b> of <i>A</i> (in <b>U</b>) is defined as </p> <dl><dd><span class="texhtml"><i>A</i><sup>C</sup> := {<i>x</i> ∈ <b>U</b> | <i>x</i> ∉ <i>A</i>}</span>.</dd></dl> <p>In other words, <i>A</i><sup>C</sup> ("<i>A-complement</i>"; sometimes simply <i>A'</i>, "<i>A-prime</i>" ) is the set of all members of <b>U</b> which are not members of <i>A</i>. Thus with <b>R</b>, <b>Z</b> and <i>O</i> defined as in the section on subsets, if <b>Z</b> is the universal set, then <i>O<sup>C</sup></i> is the set of even integers, while if <b>R</b> is the universal set, then <i>O<sup>C</sup></i> is the set of all real numbers that are either even integers or not integers at all. </p> <div class="mw-heading mw-heading2"><h2 id="Unions,_intersections,_and_relative_complements"><span id="Unions.2C_intersections.2C_and_relative_complements"></span>Unions, intersections, and relative complements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=15" title="Edit section: Unions, intersections, and relative complements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given two sets <i>A</i> and <i>B</i>, their <b><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></b> is the set consisting of all objects which are elements of <i>A</i> or of <i>B</i> or of both (see <a href="/wiki/Axiom_of_union" title="Axiom of union">axiom of union</a>). It is denoted by <span class="texhtml"><i>A</i> ∪ <i>B</i></span>. </p><p>The <b><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></b> of <i>A</i> and <i>B</i> is the set of all objects which are both in <i>A</i> and in <i>B</i>. It is denoted by <span class="texhtml"><i>A</i> ∩ <i>B</i></span>. </p><p>Finally, the <b><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">relative complement</a></b> of <i>B</i> relative to <i>A</i>, also known as the <b>set theoretic difference</b> of <i>A</i> and <i>B</i>, is the set of all objects that belong to <i>A</i> but <i>not</i> to <i>B</i>. It is written as <span class="texhtml"><i>A</i> \ <i>B</i></span> or <span class="texhtml"><i>A</i> − <i>B</i></span>. </p><p>Symbolically, these are respectively </p> <dl><dd><span class="texhtml"><i>A</i> ∪ B := {<i>x</i> | (<i>x</i> ∈ <i>A</i>) <a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a> (<i>x</i> ∈ <i>B</i>)}</span>;</dd> <dd><span class="texhtml"><i>A</i> ∩ <i>B</i> := {<i>x</i> | (<i>x</i> ∈ <i>A</i>) <a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a> (<i>x</i> ∈ <i>B</i>)} = {<i>x</i> ∈ <i>A</i> | <i>x</i> ∈ <i>B</i>} = {<i>x</i> ∈ <i>B</i> | <i>x</i> ∈ <i>A</i>}</span>;</dd> <dd><span class="texhtml"><i>A</i> \ <i>B</i> := {<i>x</i> | (<i>x</i> ∈ <i>A</i>) ∧ <a href="/wiki/Negation" title="Negation">¬</a> (<i>x</i> ∈ <i>B</i>) } = {<i>x</i> ∈ <i>A</i> | ¬ (<i>x</i> ∈ <i>B</i>)}</span>.</dd></dl> <p>The set <i>B</i> doesn't have to be a subset of <i>A</i> for <span class="texhtml"><i>A</i> \ <i>B</i></span> to make sense; this is the difference between the relative complement and the absolute complement (<span class="texhtml"><i>A</i><sup>C</sup> = <i>U</i> \ <i>A</i></span>) from the previous section. </p><p>To illustrate these ideas, let <i>A</i> be the set of left-handed people, and let <i>B</i> be the set of people with blond hair. Then <span class="texhtml"><i>A</i> ∩ <i>B</i></span> is the set of all left-handed blond-haired people, while <span class="texhtml"><i>A</i> ∪ <i>B</i></span> is the set of all people who are left-handed or blond-haired or both. <span class="texhtml"><i>A</i> \ <i>B</i></span>, on the other hand, is the set of all people that are left-handed but not blond-haired, while <span class="texhtml"><i>B</i> \ <i>A</i></span> is the set of all people who have blond hair but aren't left-handed. </p><p>Now let <i>E</i> be the set of all human beings, and let <i>F</i> be the set of all living things over 1000 years old. What is <span class="texhtml"><i>E</i> ∩ <i>F</i></span> in this case? No living human being is <a href="/wiki/Oldest_people" title="Oldest people">over 1000 years old</a>, so <span class="texhtml"><i>E</i> ∩ <i>F</i></span> must be the <a href="/wiki/Empty_set" title="Empty set">empty set</a> {}. </p><p>For any set <i>A</i>, the power set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f264d19e21604793c6dc54f8044df454db82744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}"></span> is a <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> under the operations of union and intersection. </p> <div class="mw-heading mw-heading2"><h2 id="Ordered_pairs_and_Cartesian_products">Ordered pairs and Cartesian products</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=16" title="Edit section: Ordered pairs and Cartesian products"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Intuitively, an <b><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></b> is simply a collection of two objects such that one can be distinguished as the <i>first element</i> and the other as the <i>second element</i>, and having the fundamental property that, two ordered pairs are equal if and only if their <i>first elements</i> are equal and their <i>second elements</i> are equal. </p><p>Formally, an ordered pair with <b>first coordinate</b> <i>a</i>, and <b>second coordinate</b> <i>b</i>, usually denoted by (<i>a</i>, <i>b</i>), can be defined as the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\{a\},\{a,b\}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\{a\},\{a,b\}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf616e0eb31348def4d95b4fa0832a4928196cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.147ex; height:2.843ex;" alt="{\displaystyle \{\{a\},\{a,b\}\}.}"></span> </p><p>It follows that, two ordered pairs (<i>a</i>,<i>b</i>) and (<i>c</i>,<i>d</i>) are equal if and only if <span class="texhtml"><i>a</i> = <i>c</i></span> and <span class="texhtml"><i>b</i> = <i>d</i></span>. </p><p>Alternatively, an ordered pair can be formally thought of as a set {a,b} with a <a href="/wiki/Total_order" title="Total order">total order</a>. </p><p>(The notation (<i>a</i>, <i>b</i>) is also used to denote an <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open interval</a> on the <a href="/wiki/Real_number_line" class="mw-redirect" title="Real number line">real number line</a>, but the context should make it clear which meaning is intended. Otherwise, the notation ]<i>a</i>, <i>b</i>[ may be used to denote the open interval whereas (<i>a</i>, <i>b</i>) is used for the ordered pair). </p><p>If <i>A</i> and <i>B</i> are sets, then the <b><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></b> (or simply <b>product</b>) is defined to be: </p> <dl><dd><span class="texhtml"><i>A</i> × <i>B</i> = {(<i>a</i>,<i>b</i>) | <i>a</i> ∈ <i>A</i> and <i>b</i> ∈ <i>B</i>}.</span></dd></dl> <p>That is, <span class="texhtml"><i>A</i> × <i>B</i></span> is the set of all ordered pairs whose first coordinate is an element of <i>A</i> and whose second coordinate is an element of <i>B</i>. </p><p>This definition may be extended to a set <span class="texhtml"><i>A</i> × <i>B</i> × <i>C</i></span> of ordered triples, and more generally to sets of ordered <a href="/wiki/N-tuple" class="mw-redirect" title="N-tuple">n-tuples</a> for any positive integer <i>n</i>. It is even possible to define infinite <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian products</a>, but this requires a more recondite definition of the product. </p><p>Cartesian products were first developed by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> in the context of <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>. If <b>R</b> denotes the set of all <a href="/wiki/Real_number" title="Real number">real numbers</a>, then <span class="texhtml"><b>R</b><sup>2</sup> := <b>R</b> × <b>R</b></span> represents the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> and <span class="texhtml"><b>R</b><sup>3</sup> := <b>R</b> × <b>R</b> × <b>R</b></span> represents three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Some_important_sets">Some important sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=17" title="Edit section: Some important sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are some ubiquitous sets for which the notation is almost universal. Some of these are listed below. In the list, <i>a</i>, <i>b</i>, and <i>c</i> refer to <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, and <i>r</i> and <i>s</i> are <a href="/wiki/Real_number" title="Real number">real numbers</a>. </p> <ol><li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> are used for counting. A <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> capital <b>N</b> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>) often represents this set.</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> appear as solutions for <i>x</i> in equations like <i>x</i> + <i>a</i> = <i>b</i>. A blackboard bold capital <b>Z</b> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>) often represents this set (from the German <i>Zahlen</i>, meaning <i>numbers</i>).</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> appear as solutions to equations like <i>a</i> + <i>bx</i> = <i>c</i>. A blackboard bold capital <b>Q</b> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>) often represents this set (for <i><a href="/wiki/Quotient" title="Quotient">quotient</a></i>, because R is used for the set of real numbers).</li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> appear as solutions to <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> equations (with integer coefficients) and may involve <a href="/wiki/Nth_root" title="Nth root">radicals</a> (including <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i={\sqrt {-1\,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-1\,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ec26ccf26272e80eb5b74c562c7cb910c44843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.194ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-1\,}}}"></span>) and certain other <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>. A <b>Q</b> with an overline (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377a8814b1ca454c488e409813988dd5dd906148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.923ex; height:3.343ex;" alt="{\displaystyle {\overline {\mathbb {Q} }}}"></span>) often represents this set. The overline denotes the operation of <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a>.</li> <li><a href="/wiki/Real_number" title="Real number">Real numbers</a> represent the "real line" and include all numbers that can be approximated by rationals. These numbers may be rational or algebraic but may also be <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a>, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital <b>R</b> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>) often represents this set.</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> are sums of a real and an imaginary number: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+s\,i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mspace width="thinmathspace" /> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+s\,i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da30e786ecc4da6e0952c87573e09541eb9439fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.169ex; height:2.343ex;" alt="{\displaystyle r+s\,i}"></span>. Here either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> (or both) can be zero; thus, the set of real numbers and the set of strictly imaginary numbers are subsets of the set of complex numbers, which form an <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> for the set of real numbers, meaning that every polynomial with coefficients in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> has at least one <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> in this set. A blackboard bold capital <b>C</b> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>) often represents this set. Note that since a number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+s\,i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mspace width="thinmathspace" /> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+s\,i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da30e786ecc4da6e0952c87573e09541eb9439fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.169ex; height:2.343ex;" alt="{\displaystyle r+s\,i}"></span> can be identified with a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f3e3778daed31bfa6149169529acb5d9fcc8b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle (r,s)}"></span> in the plane, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> is basically "the same" as the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb99f01c438a62e4ac5af8cff4eb402739ed67a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \times \mathbb {R} }"></span> ("the same" meaning that any point in one determines a unique point in the other and for the result of calculations, it doesn't matter which one is used for the calculation, as long as multiplication rule is appropriate for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>).</li></ol> <div class="mw-heading mw-heading2"><h2 id="Paradoxes_in_early_set_theory">Paradoxes in early set theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=18" title="Edit section: Paradoxes in early set theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Paradox" title="Paradox">Paradox</a></div> <p>The unrestricted formation principle of sets referred to as the <a href="/wiki/Axiom_schema_of_specification#Unrestricted_comprehension" title="Axiom schema of specification">axiom schema of unrestricted comprehension</a>, </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent">If <span class="texhtml"><i>P</i></span> is a property, then there exists a set <span class="texhtml"><i>Y</i> = {<i>x</i> : <i>P</i>(<i>x</i>)}</span>,<sup id="cite_ref-FOOTNOTEJech20024_14-0" class="reference"><a href="#cite_note-FOOTNOTEJech20024-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></div> <p>is the source of several early appearing paradoxes: </p> <ul><li><span class="texhtml"><var style="padding-right: 1px;">Y</var> = {<var style="padding-right: 1px;">x</var> | <var style="padding-right: 1px;">x</var> is an ordinal}</span> led, in the year 1897, to the <a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a>, the first published <a href="/wiki/Antinomy" title="Antinomy">antinomy</a>.</li> <li><span class="texhtml"><var style="padding-right: 1px;">Y</var> = {<var style="padding-right: 1px;">x</var> | <var style="padding-right: 1px;">x</var> is a cardinal}</span> produced <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">Cantor's paradox</a> in 1897.<sup id="cite_ref-Letter_to_Hilbert_8-1" class="reference"><a href="#cite_note-Letter_to_Hilbert-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li><span class="texhtml"><var style="padding-right: 1px;">Y</var> = {<var style="padding-right: 1px;">x</var> | {} = {}}</span> yielded <b>Cantor's second antinomy</b> in the year 1899.<sup id="cite_ref-Letters_to_Dedekind_10-1" class="reference"><a href="#cite_note-Letters_to_Dedekind-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Here the property <span class="texhtml mvar" style="font-style:italic;">P</span> is true for all <span class="texhtml mvar" style="font-style:italic;">x</span>, whatever <span class="texhtml mvar" style="font-style:italic;">x</span> may be, so <span class="texhtml mvar" style="font-style:italic;">Y</span> would be a <a href="/wiki/Universal_set" title="Universal set">universal set</a>, containing everything.</li> <li><span class="texhtml"><var style="padding-right: 1px;">Y</var> = {<var style="padding-right: 1px;">x</var> | <var style="padding-right: 1px;">x</var> ∉ <var style="padding-right: 1px;">x</var>}</span>, i.e. the set of all sets that do not contain themselves as elements, gave <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a> in 1902.</li></ul> <p>If the axiom schema of unrestricted comprehension is weakened to the <a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">axiom schema of specification</a> or <b>axiom schema of separation</b>, </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">If <span class="texhtml mvar" style="font-style:italic;">P</span> is a property, then for any set <span class="texhtml mvar" style="font-style:italic;">X</span> there exists a set <span class="texhtml"><var style="padding-right: 1px;">Y</var> = {<var style="padding-right: 1px;">x</var> ∈ <var style="padding-right: 1px;">X</var> : <var style="padding-right: 1px;">P</var>(<var style="padding-right: 1px;">x</var>)}</span>,<sup id="cite_ref-FOOTNOTEJech20024_14-1" class="reference"><a href="#cite_note-FOOTNOTEJech20024-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></div> <p>then all the above paradoxes disappear.<sup id="cite_ref-FOOTNOTEJech20024_14-2" class="reference"><a href="#cite_note-FOOTNOTEJech20024-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> There is a corollary. With the axiom schema of separation as an axiom of the theory, it follows, as a theorem of the theory: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">The set of all sets does not exist.</div> <p>Or, more spectacularly (Halmos' phrasing<sup id="cite_ref-FOOTNOTEHalmos1974Chapter_2_15-0" class="reference"><a href="#cite_note-FOOTNOTEHalmos1974Chapter_2-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>): There is no <a href="/wiki/Domain_of_discourse" title="Domain of discourse">universe</a>. <i>Proof</i>: Suppose that it exists and call it <span class="texhtml mvar" style="font-style:italic;">U</span>. Now apply the axiom schema of separation with <span class="texhtml"><var style="padding-right: 1px;">X</var> = <var style="padding-right: 1px;">U</var></span> and for <span class="texhtml"><var style="padding-right: 1px;">P</var>(<var style="padding-right: 1px;">x</var>)</span> use <span class="texhtml"><var style="padding-right: 1px;">x</var> ∉ <var style="padding-right: 1px;">x</var></span>. This leads to Russell's paradox again. Hence <span class="texhtml mvar" style="font-style:italic;">U</span> cannot exist in this theory.<sup id="cite_ref-FOOTNOTEJech20024_14-3" class="reference"><a href="#cite_note-FOOTNOTEJech20024-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Related to the above constructions is formation of the set </p> <ul><li><span class="texhtml"><var style="padding-right: 1px;">Y</var> = {<var style="padding-right: 1px;">x</var> | (<var style="padding-right: 1px;">x</var> ∈ <var style="padding-right: 1px;">x</var>) → {} ≠ {}}</span>,</li></ul> <p>where the statement following the implication certainly is false. It follows, from the definition of <span class="texhtml mvar" style="font-style:italic;">Y</span>, using the usual inference rules (and some afterthought when reading the proof in the linked article below) both that <span class="texhtml"><var style="padding-right: 1px;">Y</var> ∈ <var style="padding-right: 1px;">Y</var> → {} ≠ {}</span> and <span class="texhtml"><var style="padding-right: 1px;">Y</var> ∈ <var style="padding-right: 1px;">Y</var></span> holds, hence <span class="texhtml">{} ≠ {}</span>. This is <a href="/wiki/Curry%27s_paradox#Naive_set_theory" title="Curry's paradox">Curry's paradox</a>. </p><p>It is (perhaps surprisingly) not the possibility of <span class="texhtml"><var style="padding-right: 1px;">x</var> ∈ <var style="padding-right: 1px;">x</var></span> that is problematic. It is again the axiom schema of unrestricted comprehension allowing <span class="texhtml">(<var style="padding-right: 1px;">x</var> ∈ <var style="padding-right: 1px;">x</var>) → {} ≠ {}</span> for <span class="texhtml"><var style="padding-right: 1px;">P</var>(<var style="padding-right: 1px;">x</var>)</span>. With the axiom schema of specification instead of unrestricted comprehension, the conclusion <span class="texhtml"><var style="padding-right: 1px;">Y</var> ∈ <var style="padding-right: 1px;">Y</var></span> does not hold and hence <span class="texhtml">{} ≠ {}</span> is not a logical consequence. </p><p>Nonetheless, the possibility of <span class="texhtml"><var style="padding-right: 1px;">x</var> ∈ <var style="padding-right: 1px;">x</var></span> is often removed explicitly<sup id="cite_ref-FOOTNOTEHalmos1974See_discussion_around_Russell's_paradox_16-0" class="reference"><a href="#cite_note-FOOTNOTEHalmos1974See_discussion_around_Russell's_paradox-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> or, e.g. in ZFC, implicitly,<sup id="cite_ref-FOOTNOTEJech2002Section_1.6_17-0" class="reference"><a href="#cite_note-FOOTNOTEJech2002Section_1.6-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> by demanding the <a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">axiom of regularity</a> to hold.<sup id="cite_ref-FOOTNOTEJech2002Section_1.6_17-1" class="reference"><a href="#cite_note-FOOTNOTEJech2002Section_1.6-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> One consequence of it is </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">There is no set <span class="texhtml mvar" style="font-style:italic;">X</span> for which <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">X</span> ∈ <var style="padding-right: 1px;">X</var></span>,</div> <p>or, in other words, no set is an element of itself.<sup id="cite_ref-FOOTNOTEJech200261_18-0" class="reference"><a href="#cite_note-FOOTNOTEJech200261-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The axiom schema of separation is simply too weak (while unrestricted comprehension is a very strong axiom—too strong for set theory) to develop set theory with its usual operations and constructions outlined above.<sup id="cite_ref-FOOTNOTEJech20024_14-4" class="reference"><a href="#cite_note-FOOTNOTEJech20024-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The axiom of regularity is of a restrictive nature as well. Therefore, one is led to the formulation of other axioms to guarantee the existence of enough sets to form a set theory. Some of these have been described informally above and many others are possible. Not all conceivable axioms can be combined freely into consistent theories. For example, the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> of ZFC is incompatible with the conceivable "every set of reals is <a href="/wiki/Lebesgue_measurable" class="mw-redirect" title="Lebesgue measurable">Lebesgue measurable</a>". The former implies the latter is false. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output 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title="Partially ordered set">Partially ordered set</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=20" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/s.html">"Earliest Known Uses of Some of the Words of Mathematics (S)"</a>. April 14, 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics+%28S%29&rft.date=2020-04-14&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEHalmos1960''Naive_Set_Theory''-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHalmos1960''Naive_Set_Theory''_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalmos1960">Halmos 1960</a>, <i>Naive Set Theory</i>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Jeff Miller writes that <i>naive set theory</i> (as opposed to axiomatic set theory) was used occasionally in the 1940s and became an established term in the 1950s. It appears in Hermann Weyl's review of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._A._Schilpp1946" class="citation journal cs1">P. A. Schilpp, ed. (1946). "The Philosophy of Bertrand Russell". <i>American Mathematical Monthly</i>. <b>53</b> (4): 210,</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+Philosophy+of+Bertrand+Russell&rft.volume=53&rft.issue=4&rft.pages=210&rft.date=1946&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span> and in a review by Laszlo Kalmar (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaszlo_Kalmar1946" class="citation journal cs1">Laszlo Kalmar (1946). "The Paradox of Kleene and Rosser". <i>Journal of Symbolic Logic</i>. <b>11</b> (4): 136.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=The+Paradox+of+Kleene+and+Rosser&rft.volume=11&rft.issue=4&rft.pages=136&rft.date=1946&rft.au=Laszlo+Kalmar&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The term was later popularized in a book by <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>.<sup id="cite_ref-FOOTNOTEHalmos1960''Naive_Set_Theory''_2-0" class="reference"><a href="#cite_note-FOOTNOTEHalmos1960''Naive_Set_Theory''-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_Lane1971" class="citation cs2">Mac Lane, Saunders (1971), "Categorical algebra and set-theoretic foundations", <i>Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967)</i>, Providence, RI: Amer. Math. Soc., pp. 231–240, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0282791">0282791</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Categorical+algebra+and+set-theoretic+foundations&rft.btitle=Axiomatic+Set+Theory+%28Proc.+Sympos.+Pure+Math.%2C+Vol.+XIII%2C+Part+I%2C+Univ.+California%2C+Los+Angeles%2C+Calif.%2C+1967%29&rft.place=Providence%2C+RI&rft.pages=231-240&rft.pub=Amer.+Math.+Soc.&rft.date=1971&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0282791%23id-name%3DMR&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span>. "The working mathematicians usually thought in terms of a naive set theory (probably one more or less equivalent to ZF) ... a practical requirement [of any new foundational system] could be that this system could be used "naively" by mathematicians not sophisticated in foundational research" (<a rel="nofollow" class="external text" href="https://books.google.com/books?id=TVi2AwAAQBAJ&pg=PA236">p. 236</a>).</span> </li> <li id="cite_note-FOOTNOTECantor1874-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECantor1874_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCantor1874">Cantor 1874</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFFrege1893">Frege 1893</a> In Volume 2, Jena 1903. pp. 253-261 Frege discusses the antionomy in the afterword.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFPeano1889">Peano 1889</a> Axiom 52. chap. IV produces antinomies.</span> </li> <li id="cite_note-Letter_to_Hilbert-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Letter_to_Hilbert_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Letter_to_Hilbert_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Letter from Cantor to <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> on September 26, 1897, <a href="#CITEREFMeschkowskiNilson1991">Meschkowski & Nilson 1991</a> p. 388.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Letter from Cantor to <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> on August 3, 1899, <a href="#CITEREFMeschkowskiNilson1991">Meschkowski & Nilson 1991</a> p. 408.</span> </li> <li id="cite_note-Letters_to_Dedekind-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Letters_to_Dedekind_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Letters_to_Dedekind_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Letters from Cantor to <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> on August 3, 1899 and on August 30, 1899, <a href="#CITEREFZermelo1932">Zermelo 1932</a> p. 448 (System aller denkbaren Klassen) and <a href="#CITEREFMeschkowskiNilson1991">Meschkowski & Nilson 1991</a> p. 407. (There is no set of all sets.)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">F. R. Drake, <i>Set Theory: An Introduction to Large Cardinals</i> (1974). ISBN 0 444 10535 2.</span> </li> <li id="cite_note-FOOTNOTEHalmos19749-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHalmos19749_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalmos1974">Halmos 1974</a>, p. 9.</span> </li> <li id="cite_note-FOOTNOTEHalmos197410-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHalmos197410_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalmos1974">Halmos 1974</a>, p. 10.</span> </li> <li id="cite_note-FOOTNOTEJech20024-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEJech20024_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJech20024_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJech20024_14-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJech20024_14-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJech20024_14-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFJech2002">Jech 2002</a>, p. 4.</span> </li> <li id="cite_note-FOOTNOTEHalmos1974Chapter_2-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHalmos1974Chapter_2_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalmos1974">Halmos 1974</a>, Chapter 2.</span> </li> <li id="cite_note-FOOTNOTEHalmos1974See_discussion_around_Russell's_paradox-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHalmos1974See_discussion_around_Russell's_paradox_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalmos1974">Halmos 1974</a>, See discussion around Russell's paradox.</span> </li> <li id="cite_note-FOOTNOTEJech2002Section_1.6-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEJech2002Section_1.6_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJech2002Section_1.6_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFJech2002">Jech 2002</a>, Section 1.6.</span> </li> <li id="cite_note-FOOTNOTEJech200261-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJech200261_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJech2002">Jech 2002</a>, p. 61.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, N.</a>, <i>Elements of the History of Mathematics</i>, <a href="/wiki/John_D._P._Meldrum" title="John D. P. Meldrum">John Meldrum</a> (trans.), Springer-Verlag, Berlin, Germany, 1994.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCantor1874" class="citation cs2"><a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor, Georg</a> (1874), <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583">"Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"</a>, <i><a href="/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">J. Reine Angew. Math.</a></i>, <b>1874</b> (77): 258–262, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1874.77.258">10.1515/crll.1874.77.258</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124035379">124035379</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Reine+Angew.+Math.&rft.atitle=Ueber+eine+Eigenschaft+des+Inbegriffes+aller+reellen+algebraischen+Zahlen&rft.volume=1874&rft.issue=77&rft.pages=258-262&rft.date=1874&rft_id=info%3Adoi%2F10.1515%2Fcrll.1874.77.258&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124035379%23id-name%3DS2CID&rft.aulast=Cantor&rft.aufirst=Georg&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fmain%2Fdms%2Fimg%2F%3FPPN%3DGDZPPN002155583&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span>; see also <a rel="nofollow" class="external text" href="http://bolyai.cs.elte.hu/~badam/matbsc/11o/cantor1874.pdf">pdf version</a></li> <li><a href="/wiki/Keith_J._Devlin" class="mw-redirect" title="Keith J. Devlin">Devlin, K.J.</a>, <i>The Joy of Sets: Fundamentals of Contemporary Set Theory</i>, 2nd edition, Springer-Verlag, New York, NY, 1993.</li> <li>María J. Frápolli|Frápolli, María J., 1991, "Is Cantorian set theory an iterative conception of set?". <i>Modern Logic</i>, v. 1 n. 4, 1991, 302–318.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrege1893" class="citation cs2"><a href="/wiki/Gotlob_Frege" class="mw-redirect" title="Gotlob Frege">Frege, Gottlob</a> (1893), <i>Grundgesetze der Arithmetik</i>, vol. 1, Jena</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Grundgesetze+der+Arithmetik&rft.place=Jena&rft.date=1893&rft.aulast=Frege&rft.aufirst=Gottlob&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1960" class="citation book cs1"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul</a> (1960). <i><a href="/wiki/Naive_Set_Theory_(book)" title="Naive Set Theory (book)">Naive Set Theory</a></i>. Princeton, NJ: D. Van Nostrand Company.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.place=Princeton%2C+NJ&rft.pub=D.+Van+Nostrand+Company&rft.date=1960&rft.aulast=Halmos&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1974" class="citation book cs1">Halmos, Paul (1974). <i>Naive Set Theory</i> (Reprint ed.). New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90092-6" title="Special:BookSources/0-387-90092-6"><bdi>0-387-90092-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.place=New+York&rft.edition=Reprint&rft.pub=Springer-Verlag&rft.date=1974&rft.isbn=0-387-90092-6&rft.aulast=Halmos&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos2011" class="citation book cs1">Halmos, Paul (2011). <i>Naive Set Theory</i> (Paperback ed.). Mansfield Centre, CN: D. Van Nostrand Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61427-131-4" title="Special:BookSources/978-1-61427-131-4"><bdi>978-1-61427-131-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.place=Mansfield+Centre%2C+CN&rft.edition=Paperback&rft.pub=D.+Van+Nostrand+Company&rft.date=2011&rft.isbn=978-1-61427-131-4&rft.aulast=Halmos&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span></li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJech2002" class="citation book cs1"><a href="/wiki/Thomas_Jech" title="Thomas Jech">Jech, Thomas</a> (2002). <i>Set theory, third millennium edition (revised and expanded)</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-44085-2" title="Special:BookSources/3-540-44085-2"><bdi>3-540-44085-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+theory%2C+third+millennium+edition+%28revised+and+expanded%29&rft.pub=Springer&rft.date=2002&rft.isbn=3-540-44085-2&rft.aulast=Jech&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span></li> <li><a href="/wiki/John_L._Kelley" title="John L. Kelley">Kelley, J.L.</a>, <i>General Topology</i>, Van Nostrand Reinhold, New York, NY, 1955.</li> <li><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, J.</a>, <i>From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931</i>, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-32449-8" title="Special:BookSources/0-674-32449-8">0-674-32449-8</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeschkowskiNilson1991" class="citation cs2 cs1-prop-interwiki-linked-name"><a href="https://de.wikipedia.org/wiki/Herbert_Meschkowski" class="extiw" title="de:Herbert Meschkowski">Meschkowski, Herbert</a> <span class="cs1-format">[in German]</span>; Nilson, Winfried (1991), <i>Georg Cantor: Briefe. Edited by the authors.</i>, Berlin: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-50621-7" title="Special:BookSources/3-540-50621-7"><bdi>3-540-50621-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Georg+Cantor%3A+Briefe.+Edited+by+the+authors.&rft.place=Berlin&rft.pub=Springer&rft.date=1991&rft.isbn=3-540-50621-7&rft.aulast=Meschkowski&rft.aufirst=Herbert&rft.au=Nilson%2C+Winfried&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1889" class="citation cs2"><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1889), <i>Arithmetices Principies nova Methoda exposita</i>, Turin</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetices+Principies+nova+Methoda+exposita&rft.place=Turin&rft.date=1889&rft.aulast=Peano&rft.aufirst=Giuseppe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZermelo1932" class="citation cs2"><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Zermelo, Ernst</a> (1932), <i>Georg Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind. Edited by the author.</i>, Berlin: Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Georg+Cantor%3A+Gesammelte+Abhandlungen+mathematischen+und+philosophischen+Inhalts.+Mit+erl%C3%A4uternden+Anmerkungen+sowie+mit+Erg%C3%A4nzungen+aus+dem+Briefwechsel+Cantor-Dedekind.+Edited+by+the+author.&rft.place=Berlin&rft.pub=Springer&rft.date=1932&rft.aulast=Zermelo&rft.aufirst=Ernst&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANaive+set+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naive_set_theory&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Beginnings_of_set_theory/">Beginnings of set theory</a> page at St. Andrews</li> <li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/s.html">Earliest Known Uses of Some of the Words of Mathematics (S)</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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choice">global</a></li></ul></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a> <ul><li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">projective</a></li></ul></li> <li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li> <li><a href="/wiki/Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Union</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li></ul> <ul><li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> <ul><li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li> <li><a href="/wiki/Power_set" title="Power set">Power set</a></li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost" title="Almost">Almost</a></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="/wiki/Large_cardinal" title="Large cardinal">large</a>)</li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li><a href="/wiki/Constructible_universe" title="Constructible universe">Constructible universe</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li> <li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a> <ul><li><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></li> <li><a href="/wiki/Tuple" title="Tuple">tuple</a></li></ul></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Bijection" title="Bijection">One-to-one correspondence</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li> <li><a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_set" title="Amorphous set">Amorphous</a></li> <li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a> (<a href="/wiki/Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li> <li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">Filter</a> <ul><li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">base</a></li> <li><a href="/wiki/Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li> <li><a href="/wiki/Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a> (<a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li> <li><a href="/wiki/Computable_set" title="Computable set">Recursive</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Subset" title="Subset">Subset <b>·</b> Superset</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li> <li><a href="/wiki/Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li> <li><a class="mw-selflink selflink">Naive</a></li> <li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul> <ul><li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo</a> <ul><li><a href="/wiki/General_set_theory" title="General set theory">General</a></li></ul></li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> <ul><li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li></ul></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a> <ul><li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Set_theorists" title="Category:Set theorists">Set theorists</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li> <li><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a class="mw-selflink selflink">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Mathematical_object" title="Mathematical object">Mathematical object</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span 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