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id="toc-Relation_with_scientific_theories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Terminology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Terminology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Terminology</span> </div> </a> <ul id="toc-Terminology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Layout" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Layout"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Layout</span> </div> </a> <ul id="toc-Layout-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lore" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lore"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Lore</span> </div> </a> <ul id="toc-Lore-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theorems_in_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Theorems_in_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Theorems in logic</span> </div> </a> <button aria-controls="toc-Theorems_in_logic-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 Theorems in logic subsection</span> </button> <ul id="toc-Theorems_in_logic-sublist" class="vector-toc-list"> <li id="toc-Syntax_and_semantics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Syntax_and_semantics"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Syntax and semantics</span> </div> </a> <ul id="toc-Syntax_and_semantics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretation_of_a_formal_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpretation_of_a_formal_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Interpretation of a formal theorem</span> </div> </a> <ul id="toc-Interpretation_of_a_formal_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theorems_and_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theorems_and_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Theorems and theories</span> </div> </a> <ul id="toc-Theorems_and_theories-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Citations</span> </div> </a> <button aria-controls="toc-Citations-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 Citations subsection</span> </button> <ul id="toc-Citations-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</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-2"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_cited" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Works_cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Works cited</span> </div> </a> <ul id="toc-Works_cited-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">12</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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Theorem</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input 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 90 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-90" 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">90 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D1%8D" title="Теоремэ – Kabardian" lang="kbd" hreflang="kbd" data-title="Теоремэ" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target"><span>Адыгэбзэ</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A5%E1%88%AD%E1%8C%8D%E1%8C%A5" title="እርግጥ – Amharic" lang="am" hreflang="am" data-title="እርግጥ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9" title="مبرهنة – Arabic" lang="ar" hreflang="ar" data-title="مبرهنة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="উপপাদ্য – Assamese" lang="as" hreflang="as" data-title="উপপাদ্য" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teorema" title="Teorema – Asturian" lang="ast" hreflang="ast" data-title="Teorema" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Teorem" title="Teorem – Azerbaijani" lang="az" hreflang="az" data-title="Teorem" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="উপপাদ্য – Bangla" lang="bn" hreflang="bn" data-title="উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Bashkir" lang="ba" hreflang="ba" data-title="Теорема" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0" title="Тэарэма – Belarusian" lang="be" hreflang="be" data-title="Тэарэма" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0" title="Тэарэма – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэарэма" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Bulgarian" lang="bg" hreflang="bg" data-title="Теорема" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Teorema" title="Teorema – Bosnian" lang="bs" hreflang="bs" data-title="Teorema" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema" title="Teorema – Catalan" lang="ca" hreflang="ca" data-title="Teorema" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Chuvash" lang="cv" hreflang="cv" data-title="Теорема" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matematick%C3%A1_v%C4%9Bta" title="Matematická věta – Czech" lang="cs" hreflang="cs" data-title="Matematická věta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Theorem" title="Theorem – Welsh" lang="cy" hreflang="cy" data-title="Theorem" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/S%C3%A6tning_(matematik)" title="Sætning (matematik) – Danish" lang="da" hreflang="da" data-title="Sætning (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Theorem" title="Theorem – German" lang="de" hreflang="de" data-title="Theorem" 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/Teoreem" title="Teoreem – Estonian" lang="et" hreflang="et" data-title="Teoreem" 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%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1" title="Θεώρημα – Greek" lang="el" hreflang="el" data-title="Θεώρημα" 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/Teorema" title="Teorema – Spanish" lang="es" hreflang="es" data-title="Teorema" 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/Teoremo" title="Teoremo – Esperanto" lang="eo" hreflang="eo" data-title="Teoremo" 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/Teorema" title="Teorema – Basque" lang="eu" hreflang="eu" data-title="Teorema" 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%82%D8%B6%DB%8C%D9%87" title="قضیه – Persian" lang="fa" hreflang="fa" data-title="قضیه" 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%A9or%C3%A8me" title="Théorème – French" lang="fr" hreflang="fr" data-title="Théorème" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teoirim" title="Teoirim – Irish" lang="ga" hreflang="ga" data-title="Teoirim" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Te%C3%B2irim" title="Teòirim – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Teòirim" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema" title="Teorema – Galician" lang="gl" hreflang="gl" data-title="Teorema" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B0%D0%BB%D2%BB%D0%BD" title="Таалһн – Kalmyk" lang="xal" hreflang="xal" data-title="Таалһн" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EB%A6%AC" title="정리 – Korean" lang="ko" hreflang="ko" data-title="정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B9%D5%A5%D5%B8%D6%80%D5%A5%D5%B4" title="Թեորեմ – Armenian" lang="hy" hreflang="hy" data-title="Թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="प्रमेय – Hindi" lang="hi" hreflang="hi" data-title="प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pou%C4%8Dak" title="Poučak – Croatian" lang="hr" hreflang="hr" data-title="Poučak" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Teoremo" title="Teoremo – Ido" lang="io" hreflang="io" data-title="Teoremo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema" title="Teorema – Indonesian" lang="id" hreflang="id" data-title="Teorema" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Theorema" title="Theorema – Interlingua" lang="ia" hreflang="ia" data-title="Theorema" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Setning_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Setning (stærðfræði) – Icelandic" lang="is" hreflang="is" data-title="Setning (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema" title="Teorema – Italian" lang="it" hreflang="it" data-title="Teorema" 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%9E%D7%A9%D7%A4%D7%98_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="משפט (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="משפט (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/T%C3%A9or%C3%A9ma" title="Téoréma – Javanese" lang="jv" hreflang="jv" data-title="Téoréma" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%AE%E0%B3%87%E0%B2%AF" title="ಪ್ರಮೇಯ – Kannada" lang="kn" hreflang="kn" data-title="ಪ್ರಮೇಯ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%94%E1%83%9B%E1%83%90" title="თეორემა – Georgian" lang="ka" hreflang="ka" data-title="თეორემა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Kazakh" lang="kk" hreflang="kk" data-title="Теорема" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Kyrgyz" lang="ky" hreflang="ky" data-title="Теорема" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theorema" title="Theorema – Latin" lang="la" hreflang="la" data-title="Theorema" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Teor%C4%93ma" title="Teorēma – Latvian" lang="lv" hreflang="lv" data-title="Teorēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Saz_(Mathematik)" title="Saz (Mathematik) – Luxembourgish" lang="lb" hreflang="lb" data-title="Saz (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Teorema" title="Teorema – Lithuanian" lang="lt" hreflang="lt" data-title="Teorema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%A9tel" title="Tétel – Hungarian" lang="hu" hreflang="hu" data-title="Tétel" 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%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Macedonian" lang="mk" hreflang="mk" data-title="Теорема" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="प्रमेय – Marathi" lang="mr" hreflang="mr" data-title="प्रमेय" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teorem" title="Teorem – Malay" lang="ms" hreflang="ms" data-title="Teorem" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC" title="Теорем – Mongolian" lang="mn" hreflang="mn" data-title="Теорем" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_(wiskunde)" title="Stelling (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Stelling (wiskunde)" 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/%E5%AE%9A%E7%90%86" title="定理 – Japanese" lang="ja" hreflang="ja" data-title="定理" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Teorem" title="Teorem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Teorem" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Teorem" title="Teorem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Teorem" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Teor%C3%A8ma" title="Teorèma – Occitan" lang="oc" hreflang="oc" data-title="Teorèma" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Akeekkoo" title="Akeekkoo – Oromo" lang="om" hreflang="om" data-title="Akeekkoo" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Teorema" title="Teorema – Uzbek" lang="uz" hreflang="uz" data-title="Teorema" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A5%E0%A8%BF%E0%A8%93%E0%A8%B0%E0%A8%AE" title="ਥਿਓਰਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਥਿਓਰਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema" title="Teorema – Piedmontese" lang="pms" hreflang="pms" data-title="Teorema" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie" title="Twierdzenie – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema" title="Teorema – Portuguese" lang="pt" hreflang="pt" data-title="Teorema" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorem%C4%83" title="Teoremă – Romanian" lang="ro" hreflang="ro" data-title="Teoremă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Russian" lang="ru" hreflang="ru" data-title="Теорема" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Teorema" title="Teorema – Sardinian" lang="sc" hreflang="sc" data-title="Teorema" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teorema" title="Teorema – Albanian" lang="sq" hreflang="sq" data-title="Teorema" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiurema" title="Tiurema – Sicilian" lang="scn" hreflang="scn" data-title="Tiurema" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%B8%E0%B7%9A%E0%B6%BA%E0%B6%BA" title="ප්‍රමේයය – Sinhala" lang="si" hreflang="si" data-title="ප්‍රමේයය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Theorem" title="Theorem – Simple English" lang="en-simple" hreflang="en-simple" data-title="Theorem" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Teor%C3%A9ma" title="Teoréma – Slovak" lang="sk" hreflang="sk" data-title="Teoréma" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Izrek" title="Izrek – Slovenian" lang="sl" hreflang="sl" data-title="Izrek" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%D9%85" title="تیۆرم – Central Kurdish" lang="ckb" hreflang="ckb" data-title="تیۆرم" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Serbian" lang="sr" hreflang="sr" data-title="Теорема" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorem" title="Teorem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Teorem" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lause_(matematiikka)" title="Lause (matematiikka) – Finnish" lang="fi" hreflang="fi" data-title="Lause (matematiikka)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Teorem" title="Teorem – Swedish" lang="sv" hreflang="sv" data-title="Teorem" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="தேற்றம் – Tamil" lang="ta" hreflang="ta" data-title="தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Askkud" title="Askkud – Tachelhit" lang="shi" hreflang="shi" data-title="Askkud" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97" title="ทฤษฎีบท – Thai" lang="th" hreflang="th" data-title="ทฤษฎีบท" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Teorem" title="Teorem – Turkish" lang="tr" hreflang="tr" data-title="Teorem" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Теорема – Ukrainian" lang="uk" hreflang="uk" data-title="Теорема" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%DB%81" title="قضیہ – Urdu" lang="ur" hreflang="ur" data-title="قضیہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_to%C3%A1n_h%E1%BB%8Dc" title="Định lý toán học – Vietnamese" lang="vi" hreflang="vi" data-title="Định lý toán học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AE%9A%E7%90%86" title="定理 – Wu" lang="wuu" hreflang="wuu" data-title="定理" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%98%D7%A2%D7%90%D7%A8%D7%A2%D7%9D" title="טעארעם – Yiddish" lang="yi" hreflang="yi" data-title="טעארעם" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AE%9A%E7%90%86" title="定理 – Cantonese" lang="yue" hreflang="yue" data-title="定理" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%9A%E7%90%86" title="定理 – Chinese" lang="zh" hreflang="zh" data-title="定理" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B4%B0%E2%B5%99%E2%B4%BD%E2%B4%BD%E2%B5%93%E2%B4%B7" title="ⴰⵙⴽⴽⵓⴷ – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⴰⵙⴽⴽⵓⴷ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">In mathematics, a statement that has been proven</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Teorema_(disambiguation)" class="mw-disambig" title="Teorema (disambiguation)">Teorema</a>, <a href="/wiki/Theorema_(disambiguation)" class="mw-disambig" title="Theorema (disambiguation)">Theorema</a>, or <a href="/wiki/Theory_(disambiguation)" class="mw-disambig" title="Theory (disambiguation)">Theory</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pythagorean_Proof_(3).PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Pythagorean_Proof_%283%29.PNG/200px-Pythagorean_Proof_%283%29.PNG" decoding="async" width="200" height="389" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Pythagorean_Proof_%283%29.PNG/300px-Pythagorean_Proof_%283%29.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Pythagorean_Proof_%283%29.PNG/400px-Pythagorean_Proof_%283%29.PNG 2x" data-file-width="614" data-file-height="1193" /></a><figcaption>The <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> has at least 370 known proofs.<sup id="cite_ref-Loomis_1-0" class="reference"><a href="#cite_note-Loomis-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Formal_logic" class="mw-redirect" title="Formal logic">formal logic</a>, a <b>theorem</b> is a <a href="/wiki/Statement_(logic)" title="Statement (logic)">statement</a> that has been <a href="/wiki/Mathematical_proof" title="Mathematical proof">proven</a>, or can be proven.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The <i>proof</i> of a theorem is a <a href="/wiki/Logical_argument" class="mw-redirect" title="Logical argument">logical argument</a> that uses the inference rules of a <a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive system</a> to establish that the theorem is a <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a> of the <a href="/wiki/Axiom" title="Axiom">axioms</a> and previously proved theorems. </p><p>In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> with the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> (ZFC), or of a less powerful theory, such as <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as <i>theorems</i> only the most important results, and use the terms <i>lemma</i>, <i>proposition</i> and <i>corollary</i> for less important theorems. </p><p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, the concepts of theorems and proofs have been <a href="/wiki/Formal_system" title="Formal system">formalized</a> in order to allow mathematical reasoning about them. In this context, statements become <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulas</a> of some <a href="/wiki/Formal_language" title="Formal language">formal language</a>. A <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">theory</a> consists of some basis statements called <i>axioms</i>, and some <i>deducing rules</i> (sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> This formalization led to <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, which allows proving general theorems about theorems and proofs. In particular, <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> show that every <a href="/wiki/Consistency" title="Consistency">consistent</a> theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory). </p><p>As the axioms are often abstractions of properties of the <a href="/wiki/Physical_world" class="mw-redirect" title="Physical world">physical world</a>, theorems may be considered as expressing some truth, but in contrast to the notion of a <a href="/wiki/Scientific_law" title="Scientific law">scientific law</a>, which is <i><a href="/wiki/Experimental" class="mw-redirect" title="Experimental">experimental</a></i>, the justification of the truth of a theorem is purely <a href="/wiki/Deductive" class="mw-redirect" title="Deductive">deductive</a>.<sup id="cite_ref-:0_9-0" class="reference"><a href="#cite_note-:0-9"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> A <i><a href="/wiki/Conjecture" title="Conjecture">conjecture</a></i> is a tentative proposition that may evolve to become a theorem if proven true. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Theoremhood_and_truth">Theoremhood and truth</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=1" title="Edit section: Theoremhood and truth"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Until the end of the 19th century and the <a href="/wiki/Foundational_crisis_of_mathematics" class="mw-redirect" title="Foundational crisis of mathematics">foundational crisis of mathematics</a>, all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every <a href="/wiki/Natural_number" title="Natural number">natural number</a> has a successor, and that there is exactly one <a href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</a> that passes through two given distinct points. These basic properties that were considered as absolutely evident were called <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulates</a> or <a href="/wiki/Axiom" title="Axiom">axioms</a>; for example <a href="/wiki/Euclid%27s_postulates" class="mw-redirect" title="Euclid's postulates">Euclid's postulates</a>. All theorems were proved by using implicitly or explicitly these basic properties, and, because of the evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the <a href="/wiki/Interior_angle" class="mw-redirect" title="Interior angle">interior angles</a> of a <a href="/wiki/Triangle" title="Triangle">triangle</a> equals 180°, and this was considered as an undoubtable fact. </p><p>One aspect of the foundational crisis of mathematics was the discovery of <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a> that do not lead to any contradiction, although, in such geometries, the sum of the angles of a triangle is different from 180°. So, the property <i>"the sum of the angles of a triangle equals 180°"</i> is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> leads to the contradiction of <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>. This has been resolved by elaborating the rules that are allowed for manipulating sets. </p><p>This crisis has been resolved by revisiting the foundations of mathematics to make them more <a href="/wiki/Mathematical_rigor" class="mw-redirect" title="Mathematical rigor">rigorous</a>. In these new foundations, a theorem is a <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formula</a> of a <a href="/wiki/Mathematical_theory" class="mw-redirect" title="Mathematical theory">mathematical theory</a> that can be proved from the <a href="/wiki/Axiom" title="Axiom">axioms</a> and <a href="/wiki/Inference_rules" class="mw-redirect" title="Inference rules">inference rules</a> of the theory. So, the above theorem on the sum of the angles of a triangle becomes: <i>Under the axioms and inference rules of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the sum of the interior angles of a triangle equals 180°</i>. Similarly, Russell's paradox disappears because, in an axiomatized set theory, the <i>set of all sets</i> cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is <a href="/wiki/Inconsistent" class="mw-redirect" title="Inconsistent">inconsistent</a>, and every well-formed assertion, as well as its negation, is a theorem. </p><p>In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even the significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas. </p><p>An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>, and to prove theorems about them. Examples are <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a>. In particular, there are well-formed assertions than can be proved to not be a theorem of the ambient theory, although they can be proved in a wider theory. An example is <a href="/wiki/Goodstein%27s_theorem" title="Goodstein's theorem">Goodstein's theorem</a>, which can be stated in <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Epistemological_considerations">Epistemological considerations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=2" title="Edit section: Epistemological considerations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as <b>hypotheses</b> or <a href="/wiki/Premise" title="Premise">premises</a>. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a <a href="/wiki/Necessity_and_sufficiency" title="Necessity and sufficiency">necessary consequence</a> of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain <a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive systems</a>, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., <a href="/wiki/Non-classical_logic" title="Non-classical logic">non-classical logic</a>). </p><p>Although theorems can be written in a completely symbolic form (e.g., as propositions in <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a>), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. </p><p>In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way <i>why</i> it is obviously true. In some cases, one might even be able to substantiate a theorem by using a picture as its proof. </p><p>Because theorems lie at the core of mathematics, they are also central to its <a href="/wiki/Aesthetics_of_mathematics" class="mw-redirect" title="Aesthetics of mathematics">aesthetics</a>. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> is a particularly well-known example of such a theorem.<sup id="cite_ref-:1_12-0" class="reference"><a href="#cite_note-:1-12"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Informal_account_of_theorems">Informal account of theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=3" title="Edit section: Informal account of theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Logically" class="mw-redirect" title="Logically">Logically</a>, many theorems are of the form of an <a href="/wiki/Indicative_conditional" title="Indicative conditional">indicative conditional</a>: <i>If A, then B</i>. Such a theorem does not assert <i>B</i> — only that <i>B</i> is a necessary consequence of <i>A</i>. <span class="anchor" id="Hypothesis"></span><span class="anchor" id="Conclusion"></span><span class="anchor" id="Proposition"></span>In this case, <i>A</i> is called the <i>hypothesis</i> of the theorem ("hypothesis" here means something very different from a <a href="/wiki/Conjecture" title="Conjecture">conjecture</a>), and <i>B</i> the <i>conclusion</i> of the theorem. The two together (without the proof) are called the <i>proposition</i> or <i>statement</i> of the theorem (e.g. "<i>If A, then B</i>" is the <i>proposition</i>). Alternatively, <i>A</i> and <i>B</i> can be also termed the <i><a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">antecedent</a></i> and the <i><a href="/wiki/Consequent" title="Consequent">consequent</a></i>, respectively.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The theorem "If <i>n</i> is an even <a href="/wiki/Natural_number" title="Natural number">natural number</a>, then <i>n</i>/2 is a natural number" is a typical example in which the hypothesis is "<i>n</i> is an even natural number", and the conclusion is "<i>n</i>/2 is also a natural number". </p><p>In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived from the informal one. </p><p>It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called <a href="/wiki/Axiom" title="Axiom">axioms</a> or postulates. The field of mathematics known as <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a> studies formal languages, axioms and the structure of proofs. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:4CT_Non-Counterexample_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/4CT_Non-Counterexample_1.svg/201px-4CT_Non-Counterexample_1.svg.png" decoding="async" width="201" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/4CT_Non-Counterexample_1.svg/302px-4CT_Non-Counterexample_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/4CT_Non-Counterexample_1.svg/402px-4CT_Non-Counterexample_1.svg.png 2x" data-file-width="201" data-file-height="200" /></a><figcaption>A <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">planar</a> map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.</figcaption></figure> <p>Some theorems are "<a href="/wiki/Triviality_(mathematics)" title="Triviality (mathematics)">trivial</a>", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> A theorem might be simple to state and yet be deep. An excellent example is <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>,<sup id="cite_ref-:1_12-1" class="reference"><a href="#cite_note-:1-12"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and there are many other examples of simple yet deep theorems in <a href="/wiki/Number_theory" title="Number theory">number theory</a> and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, among other areas. </p><p>Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the <a href="/wiki/Kepler_conjecture" title="Kepler conjecture">Kepler conjecture</a>. Both of these theorems are only known to be true by reducing them to a computational search that is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician <a href="/wiki/Doron_Zeilberger" title="Doron Zeilberger">Doron Zeilberger</a> has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup> and hypergeometric identities.<sup id="cite_ref-FOOTNOTEPetkovsekWilfZeilberger1996_17-0" class="reference"><a href="#cite_note-FOOTNOTEPetkovsekWilfZeilberger1996-17"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (October 2010)">page needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_with_scientific_theories">Relation with scientific theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=4" title="Edit section: Relation with scientific theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Theorems in mathematics and theories in science are fundamentally different in their <a href="/wiki/Epistemology" title="Epistemology">epistemology</a>. A scientific theory cannot be proved; its key attribute is that it is <a href="/wiki/Falsifiable" class="mw-redirect" title="Falsifiable">falsifiable</a>, that is, it makes predictions about the natural world that are testable by <a href="/wiki/Experiment" title="Experiment">experiments</a>. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.<sup id="cite_ref-:0_9-1" class="reference"><a href="#cite_note-:0-9"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CollatzFractal.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CollatzFractal.png/250px-CollatzFractal.png" decoding="async" width="250" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CollatzFractal.png/500px-CollatzFractal.png 1.5x" data-file-width="996" data-file-height="597" /></a><figcaption>The <a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a>: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The result is a <a href="/wiki/Fractal" title="Fractal">fractal</a>, which (in accordance with <a href="/wiki/Universality_(dynamical_systems)" title="Universality (dynamical systems)">universality</a>) resembles the <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a>.</figcaption></figure> <p>Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs. </p><p>For example, both the <a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a> and the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The <a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a> has been verified for start values up to about 2.88 × 10<sup>18</sup>. The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> has been verified to hold for the first 10 trillion non-trivial zeroes of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">zeta function</a>. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved. </p><p>Such evidence does not constitute proof. For example, the <a href="/wiki/Mertens_conjecture" title="Mertens conjecture">Mertens conjecture</a> is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number <i>n</i> for which the Mertens function <i>M</i>(<i>n</i>) equals or exceeds the square root of <i>n</i>) is known: all numbers less than 10<sup>14</sup> have the Mertens property, and the smallest number that does not have this property is only known to be less than the <a href="/wiki/Exponential_function" title="Exponential function">exponential</a> of 1.59 × 10<sup>40</sup>, which is approximately 10 to the power 4.3 × 10<sup>39</sup>. Since the number of particles in the universe is generally considered less than 10 to the power 100 (a <a href="/wiki/Googol" title="Googol">googol</a>), there is no hope to find an explicit counterexample by <a href="/wiki/Exhaustive_search" class="mw-redirect" title="Exhaustive search">exhaustive search</a>. </p><p>The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, <a href="/wiki/Group_theory" title="Group theory">group theory</a> (see <a href="/wiki/Mathematical_theory" class="mw-redirect" title="Mathematical theory">mathematical theory</a>). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable. </p> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=5" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time. </p> <ul><li>An <i><a href="/wiki/Axiom" title="Axiom">axiom</a></i> or <i>postulate</i> is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a <i><a href="/wiki/Definition" title="Definition">definition</a></i>, which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects.<sup id="cite_ref-FOOTNOTEWentworthSmith1913[httpsarchiveorgdetailsplanegeometry00gwenpagen25_Articles_46-7]_18-0" class="reference"><a href="#cite_note-FOOTNOTEWentworthSmith1913[httpsarchiveorgdetailsplanegeometry00gwenpagen25_Articles_46-7]-18"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Historically, axioms were regarded as "<a href="/wiki/Self-evidence" title="Self-evidence">self-evident</a>"; today they are merely <i>assumed</i> to be true.</li> <li>A <i><a href="/wiki/Conjecture" title="Conjecture">conjecture</a></i> is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (for example, <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a> and <a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a>). The term <i>hypothesis</i> is also used in this sense (for example, <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>), which should not be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example <i>problem</i> when people are not sure whether the statement should be believed to be true. <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> was historically called a theorem, although, for centuries, it was only a conjecture.</li> <li>A <i>theorem</i> is a statement that has been proven to be true based on axioms and other theorems.</li> <li>A <i><a href="/wiki/Proposition" title="Proposition">proposition</a></i> is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>. In classical geometry the term "proposition" was used differently: in <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 300 BCE</span>), all theorems and geometric constructions were called "propositions" regardless of their importance.</li> <li>A <i><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">lemma</a></i> is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a <i>theorem</i>, though the term "lemma" is usually kept as part of its name (e.g. <a href="/wiki/Gauss%27s_lemma_(polynomial)" class="mw-redirect" title="Gauss's lemma (polynomial)">Gauss's lemma</a>, <a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a>, and <a href="/wiki/Fundamental_lemma_(Langlands_program)" title="Fundamental lemma (Langlands program)">the fundamental lemma</a>).</li> <li>A <i><a href="/wiki/Corollary" title="Corollary">corollary</a></i> is a proposition that follows immediately from another theorem or axiom, with little or no required proof.<sup id="cite_ref-FOOTNOTEWentworthSmith1913[httpsarchiveorgdetailsplanegeometry00gwenpagen25_Article_51]_19-0" class="reference"><a href="#cite_note-FOOTNOTEWentworthSmith1913[httpsarchiveorgdetailsplanegeometry00gwenpagen25_Article_51]-19"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> A corollary may also be a restatement of a theorem in a simpler form, or for a <a href="/wiki/Special_case" title="Special case">special case</a>: for example, the theorem "all internal angles in a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> are <a href="/wiki/Right_angle" title="Right angle">right angles</a>" has a corollary that "all internal angles in a <i><a href="/wiki/Square" title="Square">square</a></i> are <a href="/wiki/Right_angle" title="Right angle">right angles</a>" - a square being a special case of a rectangle.</li> <li>A <i><a href="/wiki/Generalization" title="Generalization">generalization</a></i> of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a <a href="/wiki/Special_case" title="Special case">special case</a> (a <i>corollary</i>).<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup></li></ul> <p>Other terms may also be used for historical or customary reasons, for example: </p> <ul><li>An <i><a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a></i> is a theorem stating an equality between two expressions, that holds for any value within its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> (e.g. <a href="/wiki/B%C3%A9zout%27s_identity" title="Bézout's identity">Bézout's identity</a> and <a href="/wiki/Vandermonde%27s_identity" title="Vandermonde's identity">Vandermonde's identity</a>).</li> <li>A <i>rule</i> is a theorem that establishes a useful formula (e.g. <a href="/wiki/Bayes%27_rule" class="mw-redirect" title="Bayes' rule">Bayes' rule</a> and <a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a>).</li> <li>A <i><a href="/wiki/Law_(mathematics)" title="Law (mathematics)">law</a></i> or <i><a href="/wiki/Principle" title="Principle">principle</a></i> is a theorem with wide applicability (e.g. the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a>, <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law">Kolmogorov's zero–one law</a>, <a href="/wiki/Harnack%27s_principle" title="Harnack's principle">Harnack's principle</a>, the <a href="/wiki/Least-upper-bound_principle" class="mw-redirect" title="Least-upper-bound principle">least-upper-bound principle</a>, and the <a href="/wiki/Pigeonhole_principle" title="Pigeonhole principle">pigeonhole principle</a>).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>g<span class="cite-bracket">]</span></a></sup></li></ul> <p>A few well-known theorems have even more idiosyncratic names, for example, the <a href="/wiki/Euclidean_division" title="Euclidean division">division algorithm</a>, <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>, and the <a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Layout">Layout</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=6" title="Edit section: Layout"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A theorem and its proof are typically laid out as follows: </p> <dl><dd><i>Theorem</i> (name of the person who proved it, along with year of discovery or publication of the proof)</dd> <dd><i>Statement of theorem (sometimes called the </i>proposition<i>)</i></dd> <dd><i>Proof</i></dd> <dd><i>Description of proof</i></dd> <dd><i>End</i></dd></dl> <p>The end of the proof may be signaled by the letters <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> (<i>quod erat demonstrandum</i>) or by one of the <a href="/wiki/Tombstone_(typography)" title="Tombstone (typography)">tombstone</a> marks, such as "□" or "∎", meaning "end of proof", introduced by <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a> following their use in magazines to mark the end of an article.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>The exact style depends on the author or publication. Many publications provide instructions or <a href="/wiki/Macro_(computer_science)" title="Macro (computer science)">macros</a> for typesetting in the <a href="/wiki/Style_guide" title="Style guide">house style</a>. </p><p>It is common for a theorem to be preceded by <a href="/wiki/Definition" title="Definition">definitions</a> describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of <i>propositions</i> or <i>lemmas</i> which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. </p><p>Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem. </p> <div class="mw-heading mw-heading2"><h2 id="Lore">Lore</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=7" title="Edit section: Lore"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It has been estimated that over a quarter of a million theorems are proved every year.<sup id="cite_ref-FOOTNOTEHoffman1998204_23-0" class="reference"><a href="#cite_note-FOOTNOTEHoffman1998204-23"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The well-known <a href="/wiki/Aphorism" title="Aphorism">aphorism</a>, <a href="https://en.wikiquote.org/wiki/Paul_Erd%C5%91s" class="extiw" title="q:Paul Erdős">"A mathematician is a device for turning coffee into theorems"</a>, is probably due to <a href="/wiki/Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Alfréd Rényi</a>, although it is often attributed to Rényi's colleague <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the <a href="/wiki/Erd%C5%91s_number" title="Erdős number">number</a> of his collaborations, and his coffee drinking.<sup id="cite_ref-FOOTNOTEHoffman19987_24-0" class="reference"><a href="#cite_note-FOOTNOTEHoffman19987-24"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> is regarded by some to be the longest proof of a theorem. It comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and several ongoing projects hope to shorten and simplify this proof.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Another theorem of this type is the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by a layman.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2020)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Theorems_in_logic">Theorems in logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=8" title="Edit section: Theorems in logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, a <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">formal theory</a> is a set of sentences within a <a href="/wiki/Formal_language" title="Formal language">formal language</a>. A sentence is a <a href="/wiki/Well-formed_formulas" class="mw-redirect" title="Well-formed formulas">well-formed formula</a> with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a>. Some accounts define a theory to be closed under the <a href="/wiki/Logical_consequence#Semantic_consequence" title="Logical consequence">semantic consequence</a> relation (<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 \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }" /></span>), while others define it to be closed under the <a href="/wiki/Logical_consequence#Syntactic_consequence" title="Logical consequence">syntactic consequence</a>, or derivability relation (<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 \vdash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c0d30cf8cb7dba179e317fcde9583d842e80f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" /></span>).<sup id="cite_ref-FOOTNOTEBoolosBurgessJeffrey2007191_26-0" class="reference"><a href="#cite_note-FOOTNOTEBoolosBurgessJeffrey2007191-26"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEChiswellHodges2007172_27-0" class="reference"><a href="#cite_note-FOOTNOTEChiswellHodges2007172-27"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEEnderton2001148_28-0" class="reference"><a href="#cite_note-FOOTNOTEEnderton2001148-28"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHedman200489_29-0" class="reference"><a href="#cite_note-FOOTNOTEHedman200489-29"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHinman2005139_30-0" class="reference"><a href="#cite_note-FOOTNOTEHinman2005139-30"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHodges199333_31-0" class="reference"><a href="#cite_note-FOOTNOTEHodges199333-31"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEJohnstone198721_32-0" class="reference"><a href="#cite_note-FOOTNOTEJohnstone198721-32"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMonk1976208_33-0" class="reference"><a href="#cite_note-FOOTNOTEMonk1976208-33"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTERautenberg201081_34-0" class="reference"><a href="#cite_note-FOOTNOTERautenberg201081-34"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEvan_Dalen1994104_35-0" class="reference"><a href="#cite_note-FOOTNOTEvan_Dalen1994104-35"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Formal_languages.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Formal_languages.svg/300px-Formal_languages.svg.png" decoding="async" width="300" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Formal_languages.svg/450px-Formal_languages.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Formal_languages.svg/600px-Formal_languages.svg.png 2x" data-file-width="250" data-file-height="230" /></a><figcaption>This diagram shows the <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntactic entities</a> that can be constructed from <a href="/wiki/Formal_language" title="Formal language">formal languages</a>. The <a href="/wiki/Symbol_(formal)" title="Symbol (formal)">symbols</a> and <a href="/wiki/String_(computer_science)" title="String (computer science)">strings of symbols</a> may be broadly divided into <a href="/wiki/Nonsense" title="Nonsense">nonsense</a> and <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulas</a>. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.</figcaption></figure> <p>For a theory to be closed under a derivability relation, it must be associated with a <a href="/wiki/Formal_system#Deductive_system" title="Formal system">deductive system</a> that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context. The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. </p><p>In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be <a href="/wiki/Soundness" title="Soundness">unsound</a> relative to a given semantics, or relative to the standard <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> of the underlying language. A theory that is <a href="/wiki/Consistency#Model_theory" title="Consistency">inconsistent</a> has all sentences as theorems. </p><p>The definition of theorems as sentences of a formal language is useful within <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in <a href="/wiki/Model_theory" title="Model theory">model theory</a>, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a>. </p><p>Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement <i>about</i> a formal system (as opposed to <i>within</i> a formal system) is called a <i><a href="/wiki/Metatheorem" title="Metatheorem">metatheorem</a></i>. </p><p>Some important theorems in mathematical logic are: </p> <ul><li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness of first-order logic</a></li> <li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Completeness of first-order logic</a></li> <li><a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems of first-order arithmetic</a></li> <li><a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">Consistency of first-order arithmetic</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability theorem</a></li> <li><a href="/wiki/Entscheidungsproblem#Negative_answer" title="Entscheidungsproblem">Church-Turing theorem of undecidability</a></li> <li><a href="/wiki/L%C3%B6b%27s_theorem" title="Löb's theorem">Löb's theorem</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's theorem</a></li> <li><a href="/wiki/Craig%27s_theorem" title="Craig's theorem">Craig's theorem</a></li> <li><a href="/wiki/Cut-elimination_theorem" title="Cut-elimination theorem">Cut-elimination theorem</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Syntax_and_semantics">Syntax and semantics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=9" title="Edit section: Syntax and semantics"><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 articles: <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax (logic)</a> and <a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics (logic)</a></div> <p>The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a <i>true proposition,</i> which introduces <a href="/wiki/Semantics" title="Semantics">semantics</a>. Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. <a href="/wiki/Belief" title="Belief">belief</a>, <a href="/wiki/Theory_of_justification" class="mw-redirect" title="Theory of justification">justification</a> or other <a href="/wiki/Modal_logic" title="Modal logic">modalities</a>). The <a href="/wiki/Soundness" title="Soundness">soundness</a> of a formal system depends on whether or not all of its theorems are also <a href="/wiki/Validity_(logic)" title="Validity (logic)">validities</a>. A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a>). A formal system is considered <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">semantically complete</a> when all of its theorems are also tautologies. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretation_of_a_formal_theorem">Interpretation of a formal theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=10" title="Edit section: Interpretation of a formal theorem"><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/Interpretation_(logic)" title="Interpretation (logic)">Interpretation (logic)</a></div> <div class="mw-heading mw-heading3"><h3 id="Theorems_and_theories">Theorems and theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=11" title="Edit section: Theorems and theories"><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 articles: <a href="/wiki/Theory" title="Theory">Theory</a> and <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory (mathematical logic)</a></div> <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=Theorem&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.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 .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" /></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></span></li><li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Law_(mathematics)" title="Law (mathematics)">Law (mathematics)</a></li> <li><a href="/wiki/List_of_theorems" title="List of theorems">List of theorems</a></li> <li><a href="/wiki/List_of_theorems_called_fundamental" title="List of theorems called fundamental">List of theorems called fundamental</a></li> <li><a href="/wiki/Formula" title="Formula">Formula</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Toy_theorem" title="Toy theorem">Toy theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=13" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=14" 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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">In general, the distinction is weak, as the standard way to prove that a statement is provable consists of proving it. However, in mathematical logic, one considers often the set of all theorems of a theory, although one cannot prove them individually.</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">An exception is the original <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof of Fermat's Last Theorem</a>, which relies implicitly on <a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck universes</a>, whose existence requires the addition of a new axiom to set theory.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This reliance on a new axiom of set theory has since been removed.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Nevertheless, it is rather astonishing that the first proof of a statement expressed in elementary <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> involves the existence of very large infinite sets.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">A theory is often identified with the set of its theorems. This is avoided here for clarity, and also for not depending on <a href="/wiki/Set_theory" title="Set theory">set theory</a>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">However, both theorems and scientific law are the result of investigations. See <a href="#CITEREFHeath1897">Heath 1897</a>, p. clxxxii, Introduction, The terminology of <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>: "theorem (θεὼρνμα) from θεωρεἳν to investigate"</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Such as the derivation of the formula 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 \tan(\alpha +\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\alpha +\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6d0bb50140d08a2645192eba48649781fdbf80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.829ex; height:2.843ex;" alt="{\displaystyle \tan(\alpha +\beta )}" /></span> from the <a href="/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities" title="List of trigonometric identities">addition formulas of sine and cosine</a>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Often, when the less general or "corollary"-like theorem is proven first, it is because the proof of the more general form requires the simpler, corollary-like form, for use as a what is functionally a lemma, or "helper" theorem.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">The word <i>law</i> can also refer to an axiom, a <a href="/wiki/Rule_of_inference" title="Rule of inference">rule of inference</a>, or, in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="References">References</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Loomis-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Loomis_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFElisha_Scott_Loomis" class="citation web cs1">Elisha Scott Loomis. <a rel="nofollow" class="external text" href="http://www.eric.ed.gov/PDFS/ED037335.pdf">"The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Education_Resources_Information_Center" title="Education Resources Information Center">Education Resources Information Center</a></i>. <a href="/wiki/Institute_of_Education_Sciences" title="Institute of Education Sciences">Institute of Education Sciences</a> (IES) of the <a href="/wiki/U.S._Department_of_Education" class="mw-redirect" title="U.S. Department of Education">U.S. Department of Education</a><span class="reference-accessdate">. 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Retrieved <span class="nowrap">2 November</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=jeff560.tripod.com&rft.atitle=Earliest+Uses+of+Symbols+of+Set+Theory+and+Logic&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fset.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEHoffman1998204-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHoffman1998204_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHoffman1998">Hoffman 1998</a>, p. 204.</span> </li> <li id="cite_note-FOOTNOTEHoffman19987-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHoffman19987_24-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHoffman1998">Hoffman 1998</a>, p. 7.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://plus.maths.org/issue41/features/elwes/index.html">An enormous theorem: the classification of finite simple groups</a>, Richard Elwes, Plus Magazine, Issue 41 December 2006.</span> </li> <li id="cite_note-FOOTNOTEBoolosBurgessJeffrey2007191-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoolosBurgessJeffrey2007191_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoolosBurgessJeffrey2007">Boolos, Burgess & Jeffrey 2007</a>, p. 191.</span> </li> <li id="cite_note-FOOTNOTEChiswellHodges2007172-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEChiswellHodges2007172_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFChiswellHodges2007">Chiswell & Hodges 2007</a>, p. 172.</span> </li> <li id="cite_note-FOOTNOTEEnderton2001148-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEnderton2001148_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEnderton2001">Enderton 2001</a>, p. 148.</span> </li> <li id="cite_note-FOOTNOTEHedman200489-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHedman200489_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHedman2004">Hedman 2004</a>, p. 89.</span> </li> <li id="cite_note-FOOTNOTEHinman2005139-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHinman2005139_30-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHinman2005">Hinman 2005</a>, p. 139.</span> </li> <li id="cite_note-FOOTNOTEHodges199333-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHodges199333_31-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHodges1993">Hodges 1993</a>, p. 33.</span> </li> <li id="cite_note-FOOTNOTEJohnstone198721-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJohnstone198721_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJohnstone1987">Johnstone 1987</a>, p. 21.</span> </li> <li id="cite_note-FOOTNOTEMonk1976208-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMonk1976208_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMonk1976">Monk 1976</a>, p. 208.</span> </li> <li id="cite_note-FOOTNOTERautenberg201081-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERautenberg201081_34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRautenberg2010">Rautenberg 2010</a>, p. 81.</span> </li> <li id="cite_note-FOOTNOTEvan_Dalen1994104-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvan_Dalen1994104_35-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvan_Dalen1994">van Dalen 1994</a>, p. 104.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=16" title="Edit section: Works cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBoolosBurgessJeffrey2007" class="citation book cs1"><a href="/wiki/George_Boolos" title="George Boolos">Boolos, George</a>; <a href="/wiki/John_P._Burgess" title="John P. Burgess">Burgess, John</a>; Jeffrey, Richard (2007). <i>Computability and Logic</i> (5th ed.). Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+and+Logic&rft.edition=5th&rft.pub=Cambridge+University+Press&rft.date=2007&rft.aulast=Boolos&rft.aufirst=George&rft.au=Burgess%2C+John&rft.au=Jeffrey%2C+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEnderton2001" class="citation book cs1"><a href="/wiki/Herbert_Enderton" title="Herbert Enderton">Enderton, Herbert</a> (2001). <i>A Mathematical Introduction to Logic</i> (2nd ed.). Harcourt Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematical+Introduction+to+Logic&rft.edition=2nd&rft.pub=Harcourt+Academic+Press&rft.date=2001&rft.aulast=Enderton&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1897" class="citation book cs1"><a href="/wiki/Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Heath, Sir Thomas Little</a> (1897). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FIY5AAAAMAAJ&q=theorem"><i>The works of Archimedes</i></a>. Dover<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+works+of+Archimedes&rft.pub=Dover&rft.date=1897&rft.aulast=Heath&rft.aufirst=Sir+Thomas+Little&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFIY5AAAAMAAJ%26q%3Dtheorem&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHedman2004" class="citation book cs1">Hedman, Shawn (2004). <i>A First Course in Logic</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Logic&rft.pub=Oxford+University+Press&rft.date=2004&rft.aulast=Hedman&rft.aufirst=Shawn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHinman2005" class="citation book cs1">Hinman, Peter (2005). <i>Fundamentals of Mathematical Logic</i>. Wellesley, MA: A K Peters.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Mathematical+Logic&rft.place=Wellesley%2C+MA&rft.pub=A+K+Peters&rft.date=2005&rft.aulast=Hinman&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHoffman1998" class="citation book cs1"><a href="/wiki/Paul_Hoffman_(science_writer)" title="Paul Hoffman (science writer)">Hoffman, Paul</a> (1998). <i><a href="/wiki/The_Man_Who_Loved_Only_Numbers" title="The Man Who Loved Only Numbers">The Man Who Loved Only Numbers</a>: The Story of Paul Erdős and the Search for Mathematical Truth</i>. Hyperion, New York. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-85702-829-5" title="Special:BookSources/1-85702-829-5"><bdi>1-85702-829-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Man+Who+Loved+Only+Numbers%3A+The+Story+of+Paul+Erd%C5%91s+and+the+Search+for+Mathematical+Truth&rft.pub=Hyperion%2C+New+York&rft.date=1998&rft.isbn=1-85702-829-5&rft.aulast=Hoffman&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHodges1993" class="citation book cs1"><a href="/wiki/Wilfrid_Hodges" title="Wilfrid Hodges">Hodges, Wilfrid</a> (1993). <i>Model Theory</i>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Model+Theory&rft.pub=Cambridge+University+Press&rft.date=1993&rft.aulast=Hodges&rft.aufirst=Wilfrid&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJohnstone1987" class="citation book cs1">Johnstone, P. T. (1987). <i>Notes on Logic and Set Theory</i>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+Logic+and+Set+Theory&rft.pub=Cambridge+University+Press&rft.date=1987&rft.aulast=Johnstone&rft.aufirst=P.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMonk1976" class="citation book cs1">Monk, J. Donald (1976). <i>Mathematical Logic</i>. Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Logic&rft.pub=Springer-Verlag&rft.date=1976&rft.aulast=Monk&rft.aufirst=J.+Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPetkovsekWilfZeilberger1996" class="citation book cs1">Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). <a rel="nofollow" class="external text" href="https://dissertationeditinghelp.com/aeqb/"><i>A = B</i></a>. A.K. Peters, Wellesley, Massachusetts. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-063-6" title="Special:BookSources/1-56881-063-6"><bdi>1-56881-063-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+%3D+B&rft.pub=A.K.+Peters%2C+Wellesley%2C+Massachusetts&rft.date=1996&rft.isbn=1-56881-063-6&rft.aulast=Petkovsek&rft.aufirst=Marko&rft.au=Wilf%2C+Herbert&rft.au=Zeilberger%2C+Doron&rft_id=https%3A%2F%2Fdissertationeditinghelp.com%2Faeqb%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRautenberg2010" class="citation book cs1">Rautenberg, Wolfgang (2010). <i>A Concise Introduction to Mathematical Logic</i> (3rd ed.). Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concise+Introduction+to+Mathematical+Logic&rft.edition=3rd&rft.pub=Springer&rft.date=2010&rft.aulast=Rautenberg&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvan_Dalen1994" class="citation book cs1">van Dalen, Dirk (1994). <i>Logic and Structure</i> (3rd ed.). Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+and+Structure&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=1994&rft.aulast=van+Dalen&rft.aufirst=Dirk&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWentworthSmith1913" class="citation book cs1">Wentworth, G.; Smith, D.E. (1913). <a rel="nofollow" class="external text" href="https://archive.org/details/planegeometry00gwen"><i>Plane Geometry</i></a>. Ginn & Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plane+Geometry&rft.pub=Ginn+%26+Co.&rft.date=1913&rft.aulast=Wentworth&rft.aufirst=G.&rft.au=Smith%2C+D.E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fplanegeometry00gwen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theorem&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChiswellHodges2007" class="citation book cs1">Chiswell, Ian; Hodges, Wilfred (2007). <i>Mathematical Logic</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Logic&rft.pub=Oxford+University+Press&rft.date=2007&rft.aulast=Chiswell&rft.aufirst=Ian&rft.au=Hodges%2C+Wilfred&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHunter1996" class="citation book cs1"><a href="/wiki/Geoffrey_Hunter_(logician)" title="Geoffrey Hunter (logician)">Hunter, Geoffrey</a> (1996) [1971]. <i>Metalogic: An Introduction to the Metatheory of Standard First-Order Logic</i>. University of California Press (published 1973). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780520023567" title="Special:BookSources/9780520023567"><bdi>9780520023567</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36312727">36312727</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metalogic%3A+An+Introduction+to+the+Metatheory+of+Standard+First-Order+Logic&rft.pub=University+of+California+Press&rft.date=1996&rft_id=info%3Aoclcnum%2F36312727&rft.isbn=9780520023567&rft.aulast=Hunter&rft.aufirst=Geoffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span> (<a rel="nofollow" class="external text" href="https://archive.org/details/metalogicintrodu0000hunt">accessible to patrons with print disabilities</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMates1972" class="citation book cs1"><a href="/wiki/Benson_Mates" title="Benson Mates">Mates, Benson</a> (1972). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/elementarylogic00mate"><i>Elementary Logic</i></a></span>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-501491-X" title="Special:BookSources/0-19-501491-X"><bdi>0-19-501491-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Logic&rft.pub=Oxford+University+Press&rft.date=1972&rft.isbn=0-19-501491-X&rft.aulast=Mates&rft.aufirst=Benson&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementarylogic00mate&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" 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=Theorem&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/120px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/theorem" class="extiw" title="wiktionary:theorem">theorem</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Theorems" class="extiw" title="commons:Category:Theorems">Theorems</a> at Wikimedia Commons</li> <li><span class="citation mathworld" id="Reference-Mathworld-Theorem"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Theorem.html">"Theorem"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Theorem&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheorem" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.theoremoftheday.org/">Theorem of the Day</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 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href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/120px-Venn1001.svg.png" decoding="async" width="90" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/250px-Venn1001.svg.png 1.5x" data-file-width="384" data-file-height="280" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formal:</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/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a class="mw-selflink selflink">Theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Negation </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><b>⊥</b> <a href="/wiki/False_(logic)" title="False (logic)">False</a></li> <li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a></li> <li><a href="/wiki/Consistency" title="Consistency">Inconsistency</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_logic326" 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_logic326" 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 class="mw-selflink selflink">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="Traditional95" 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)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size: 85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" class="mw-redirect" 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)" class="mw-redirect" 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 class="mw-selflink selflink">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 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1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 5,\n [\"Anchor\"] = 1,\n [\"Authority control\"] = 1,\n [\"Circa\"] = 1,\n [\"Citation\"] = 1,\n [\"Citation needed\"] = 1,\n [\"Cite Merriam-Webster\"] = 1,\n [\"Cite book\"] = 15,\n [\"Cite journal\"] = 2,\n [\"Cite web\"] = 6,\n [\"Clear\"] = 1,\n [\"Commons category-inline\"] = 1,\n [\"Distinguish\"] = 1,\n [\"Efn\"] = 7,\n [\"Harvnb\"] = 1,\n [\"Hunter 1996\"] = 1,\n [\"Logical truth\"] = 1,\n [\"Main\"] = 3,\n [\"MathWorld\"] = 2,\n [\"Mathematical logic\"] = 1,\n [\"Mathworld\"] = 1,\n [\"Notelist\"] = 1,\n [\"Page needed\"] = 1,\n [\"Portal\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfn\"] = 15,\n [\"Short description\"] = 1,\n [\"Wiktionary\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-7dbbdd594f-d66bq","timestamp":"20250405211607","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Theorem","url":"https:\/\/en.wikipedia.org\/wiki\/Theorem","sameAs":"http:\/\/www.wikidata.org\/entity\/Q65943","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q65943","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-10-24T01:40:08Z","dateModified":"2025-04-04T00:49:48Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/1\/16\/Pythagorean_Proof_%283%29.PNG","headline":"in mathematics, a statement that has been proved"}</script> </body> </html>