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id="toc-Modern_development-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_sciences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_sciences"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Other sciences</span> </div> </a> <ul id="toc-Other_sciences-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mathematical logic</span> </div> </a> <button aria-controls="toc-Mathematical_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 Mathematical logic subsection</span> </button> <ul id="toc-Mathematical_logic-sublist" class="vector-toc-list"> <li id="toc-Logical_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logical_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Logical axioms</span> </div> </a> <ul id="toc-Logical_axioms-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Propositional_logic" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Propositional_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1.1</span> <span>Propositional logic</span> </div> </a> <ul id="toc-Propositional_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First-order_logic" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#First-order_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1.2</span> <span>First-order logic</span> </div> </a> <ul id="toc-First-order_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Non-logical_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-logical_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Non-logical axioms</span> </div> </a> <ul id="toc-Non-logical_axioms-sublist" class="vector-toc-list"> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> <li id="toc-Arithmetic" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.1</span> <span>Arithmetic</span> </div> </a> <ul id="toc-Arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_geometry" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Euclidean_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.2</span> <span>Euclidean geometry</span> </div> </a> <ul id="toc-Euclidean_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_analysis" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Real_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.3</span> <span>Real analysis</span> </div> </a> <ul id="toc-Real_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Role_in_mathematical_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Role_in_mathematical_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span><span>Role in mathematical logic</span></span> </div> </a> <ul id="toc-Role_in_mathematical_logic-sublist" class="vector-toc-list"> <li id="toc-Deductive_systems_and_completeness" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Deductive_systems_and_completeness"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Deductive systems and completeness</span> </div> </a> <ul id="toc-Deductive_systems_and_completeness-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_discussion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_discussion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Further discussion</span> </div> </a> <ul id="toc-Further_discussion-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">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </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">7</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">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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">Axiom</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 105 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-105" 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">105 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D1%8D" title="Аксиомэ – Kabardian" lang="kbd" hreflang="kbd" data-title="Аксиомэ" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target"><span>Адыгэбзэ</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Axiom" title="Axiom – Alemannic" lang="gsw" hreflang="gsw" data-title="Axiom" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A5%E1%88%99%E1%8A%95" title="እሙን – Amharic" lang="am" hreflang="am" data-title="እሙን" 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%B3%D9%84%D9%85%D8%A9_(%D9%81%D9%84%D8%B3%D9%81%D8%A9)" title="مسلمة (فلسفة) – Arabic" lang="ar" hreflang="ar" data-title="مسلمة (فلسفة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Axioma" title="Axioma – Aragonese" lang="an" hreflang="an" data-title="Axioma" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Axoma" title="Axoma – Asturian" lang="ast" hreflang="ast" data-title="Axoma" 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/Aksiom" title="Aksiom – Azerbaijani" lang="az" hreflang="az" data-title="Aksiom" 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%B8%E0%A7%8D%E0%A6%AC%E0%A6%A4%E0%A6%83%E0%A6%B8%E0%A6%BF%E0%A6%A6%E0%A7%8D%E0%A6%A7" title="স্বতঃসিদ্ধ – Bangla" lang="bn" hreflang="bn" data-title="স্বতঃসিদ্ধ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kong-siat" title="Kong-siat – Minnan" lang="nan" hreflang="nan" data-title="Kong-siat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Bashkir" lang="ba" hreflang="ba" data-title="Аксиома" 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%90%D0%BA%D1%81%D1%96%D1%91%D0%BC%D0%B0" title="Аксіёма – Belarusian" lang="be" hreflang="be" data-title="Аксіёма" 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%90%D0%BA%D1%81%D1%96%D1%91%D0%BC%D0%B0" title="Аксіёма – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Аксіёма" 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Bulgarian" lang="bg" hreflang="bg" data-title="Аксиома" 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/Aksiom" title="Aksiom – Bosnian" lang="bs" hreflang="bs" data-title="Aksiom" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Aksiom" title="Aksiom – Breton" lang="br" hreflang="br" data-title="Aksiom" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Axioma" title="Axioma – Catalan" lang="ca" hreflang="ca" data-title="Axioma" 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Chuvash" lang="cv" hreflang="cv" data-title="Аксиома" 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/Axiom" title="Axiom – Czech" lang="cs" hreflang="cs" data-title="Axiom" 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/Gwireb_(mathemateg)" title="Gwireb (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Gwireb (mathemateg)" 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/Aksiom" title="Aksiom – Danish" lang="da" hreflang="da" data-title="Aksiom" 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/Axiom" title="Axiom – German" lang="de" hreflang="de" data-title="Axiom" 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/Aksioom" title="Aksioom – Estonian" lang="et" hreflang="et" data-title="Aksioom" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BE%CE%AF%CF%89%CE%BC%CE%B1" title="Αξίωμα – Greek" lang="el" hreflang="el" data-title="Αξίωμα" 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/Axioma" title="Axioma – Spanish" lang="es" hreflang="es" data-title="Axioma" 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/Aksiomo" title="Aksiomo – Esperanto" lang="eo" hreflang="eo" data-title="Aksiomo" 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/Axioma" title="Axioma – Basque" lang="eu" hreflang="eu" data-title="Axioma" 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/%D8%A7%D8%B5%D9%84_%D9%85%D9%88%D8%B6%D9%88%D8%B9" title="اصل موضوع – Persian" lang="fa" hreflang="fa" data-title="اصل موضوع" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Axiom" title="Axiom – Fiji Hindi" lang="hif" hreflang="hif" data-title="Axiom" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Axiome" title="Axiome – French" lang="fr" hreflang="fr" data-title="Axiome" 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/Aics%C3%ADm" title="Aicsím – Irish" lang="ga" hreflang="ga" data-title="Aicsím" 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/Aicseam" title="Aicseam – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Aicseam" 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/Axioma" title="Axioma – Galician" lang="gl" hreflang="gl" data-title="Axioma" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Kabir%C5%A9_(axiom)" title="Kabirũ (axiom) – Kikuyu" lang="ki" hreflang="ki" data-title="Kabirũ (axiom)" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%B5%EB%A6%AC" title="공리 – Korean" lang="ko" hreflang="ko" data-title="공리" 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%B1%D6%84%D5%BD%D5%AB%D5%B8%D5%B4" title="Աքսիոմ – Armenian" lang="hy" hreflang="hy" data-title="Աքսիոմ" 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%85%E0%A4%AD%E0%A4%BF%E0%A4%97%E0%A5%83%E0%A4%B9%E0%A5%80%E0%A4%A4" title="अभिगृहीत – Hindi" lang="hi" hreflang="hi" data-title="अभिगृहीत" 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/Aksiom" title="Aksiom – Croatian" lang="hr" hreflang="hr" data-title="Aksiom" 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/Axiomo" title="Axiomo – Ido" lang="io" hreflang="io" data-title="Axiomo" 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/Aksioma" title="Aksioma – Indonesian" lang="id" hreflang="id" data-title="Aksioma" 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/Axioma" title="Axioma – Interlingua" lang="ia" hreflang="ia" data-title="Axioma" 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/Frumsenda" title="Frumsenda – Icelandic" lang="is" hreflang="is" data-title="Frumsenda" 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/Assioma_(matematica)" title="Assioma (matematica) – Italian" lang="it" hreflang="it" data-title="Assioma (matematica)" 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%90%D7%A7%D7%A1%D7%99%D7%95%D7%9E%D7%94" title="אקסיומה – Hebrew" lang="he" hreflang="he" data-title="אקסיומה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%A5%E1%83%A1%E1%83%98%E1%83%9D%E1%83%9B%E1%83%90" title="აქსიომა – Georgian" lang="ka" hreflang="ka" data-title="აქსიომა" 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Kazakh" lang="kk" hreflang="kk" data-title="Аксиома" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Aksy%C3%B2m" title="Aksyòm – Haitian Creole" lang="ht" hreflang="ht" data-title="Aksyòm" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Kyrgyz" lang="ky" hreflang="ky" data-title="Аксиома" 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/Axioma" title="Axioma – Latin" lang="la" hreflang="la" data-title="Axioma" 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/Aksioma" title="Aksioma – Latvian" lang="lv" hreflang="lv" data-title="Aksioma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Aksioma" title="Aksioma – Lithuanian" lang="lt" hreflang="lt" data-title="Aksioma" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Axioma" title="Axioma – Limburgish" lang="li" hreflang="li" data-title="Axioma" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Axi%C3%B3ma" title="Axióma – Hungarian" lang="hu" hreflang="hu" data-title="Axióma" 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Macedonian" lang="mk" hreflang="mk" data-title="Аксиома" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Matoan-kevitra" title="Matoan-kevitra – Malagasy" lang="mg" hreflang="mg" data-title="Matoan-kevitra" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B5%8D%E0%B4%B5%E0%B4%AF%E0%B4%82%E2%80%8C%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%BE%E0%B4%A3%E0%B4%82" title="സ്വയം‌സിദ്ധപ്രമാണം – Malayalam" lang="ml" hreflang="ml" data-title="സ്വയം‌സിദ്ധപ്രമാണം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A8%D8%AF%D9%8A%D9%87%D9%8A%D9%87" title="بديهيه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="بديهيه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Aksiom" title="Aksiom – Malay" lang="ms" hreflang="ms" data-title="Aksiom" 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC" title="Аксиом – Mongolian" lang="mn" hreflang="mn" data-title="Аксиом" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%99%E1%80%BE%E1%80%90%E1%80%BA%E1%80%9B%E1%80%8A%E1%80%BA%E1%80%81%E1%80%BB%E1%80%80%E1%80%BA" title="မှတ်ရည်ချက် – Burmese" lang="my" hreflang="my" data-title="မှတ်ရည်ချက်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Axioma" title="Axioma – Dutch" lang="nl" hreflang="nl" data-title="Axioma" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%8F%E0%A4%95%E0%A5%8D%E0%A4%9C%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="एक्जियम – Newari" lang="new" hreflang="new" data-title="एक्जियम" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%85%AC%E7%90%86" title="公理 – Japanese" lang="ja" hreflang="ja" data-title="公理" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Aksiom" title="Aksiom – Northern Frisian" lang="frr" hreflang="frr" data-title="Aksiom" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Aksiom" title="Aksiom – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Aksiom" 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/Aksiom" title="Aksiom – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Aksiom" 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-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Axiome" title="Axiome – Novial" lang="nov" hreflang="nov" data-title="Axiome" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Axi%C3%B2ma" title="Axiòma – Occitan" lang="oc" hreflang="oc" data-title="Axiòma" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%BE" title="Аксиомо – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Аксиомо" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Yaad-yaboo" title="Yaad-yaboo – Oromo" lang="om" hreflang="om" data-title="Yaad-yaboo" 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/Aksioma" title="Aksioma – Uzbek" lang="uz" hreflang="uz" data-title="Aksioma" 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%A4%E0%A8%A4%E0%A8%B8%E0%A8%AE%E0%A8%95" title="ਤਤਸਮਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਤਤਸਮਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D9%86%DB%8C%D8%A7_%D9%BE%D8%B1%D9%85%D9%86%DB%8C%D8%A7" title="منیا پرمنیا – Western Punjabi" lang="pnb" hreflang="pnb" data-title="منیا پرمنیا" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Aksjomat" title="Aksjomat – Polish" lang="pl" hreflang="pl" data-title="Aksjomat" 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/Axioma" title="Axioma – Portuguese" lang="pt" hreflang="pt" data-title="Axioma" 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/Axiom%C4%83" title="Axiomă – Romanian" lang="ro" hreflang="ro" data-title="Axiomă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D1%96%D0%BE%D0%BC%D0%B0" title="Аксіома – Rusyn" lang="rue" hreflang="rue" data-title="Аксіома" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Russian" lang="ru" hreflang="ru" data-title="Аксиома" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Yakut" lang="sah" hreflang="sah" data-title="Аксиома" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Define:Axiom" title="Define:Axiom – Scots" lang="sco" hreflang="sco" data-title="Define:Axiom" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Aksioma" title="Aksioma – Albanian" lang="sq" hreflang="sq" data-title="Aksioma" 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/Assioma" title="Assioma – Sicilian" lang="scn" hreflang="scn" data-title="Assioma" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Axiom" title="Axiom – Simple English" lang="en-simple" hreflang="en-simple" data-title="Axiom" 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/Axi%C3%B3ma" title="Axióma – Slovak" lang="sk" hreflang="sk" data-title="Axióma" 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/Aksiom" title="Aksiom – Slovenian" lang="sl" hreflang="sl" data-title="Aksiom" 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%A8%DB%95%DA%B5%DA%AF%DB%95%D9%86%DB%95%D9%88%DB%8C%D8%B3%D8%AA" title="بەڵگەنەویست – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بەڵگەنەویست" 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома – Serbian" lang="sr" hreflang="sr" data-title="Аксиома" 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/Aksiom" title="Aksiom – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Aksiom" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Aksioma" title="Aksioma – Sundanese" lang="su" hreflang="su" data-title="Aksioma" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aksiooma" title="Aksiooma – Finnish" lang="fi" hreflang="fi" data-title="Aksiooma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://sv.wikipedia.org/wiki/Axiom" title="Axiom – Swedish" lang="sv" hreflang="sv" data-title="Axiom" data-language-autonym="Svenska" 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searchaux" style="display:none">Statement that is taken to be true</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/Axion" title="Axion">axion</a> or <a href="/wiki/Axon" title="Axon">axon</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Several terms redirect here. For other uses, see <a href="/wiki/Axiom_(disambiguation)" class="mw-disambig" title="Axiom (disambiguation)">Axiom (disambiguation)</a>, <a href="/wiki/Axiomatic_(disambiguation)" class="mw-disambig" title="Axiomatic (disambiguation)">Axiomatic (disambiguation)</a>, and <a href="/wiki/Postulation_(algebraic_geometry)" class="mw-redirect" title="Postulation (algebraic geometry)">Postulation (algebraic geometry)</a>.</div> <p class="mw-empty-elt"> </p><p>An <b>axiom</b>, <b>postulate</b>, or <b>assumption</b> is a <a href="/wiki/Statement_(logic)" title="Statement (logic)">statement</a> that is taken to be <a href="/wiki/Truth" title="Truth">true</a>, to serve as a <a href="/wiki/Premise" title="Premise">premise</a> or starting point for further reasoning and arguments. The word comes from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a> word <span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BC%80%CE%BE%CE%AF%CF%89%CE%BC%CE%B1#Ancient_Greek" class="extiw" title="wikt:ἀξίωμα">ἀξίωμα</a></span> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>axíōma</i></span>), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The precise <a href="/wiki/Definition" title="Definition">definition</a> varies across fields of study. In <a href="/wiki/Classic_philosophy" class="mw-redirect" title="Classic philosophy">classic philosophy</a>, an axiom is a statement that is so <a href="/wiki/Self-evidence" title="Self-evidence">evident</a> or well-established, that it is accepted without controversy or question.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In modern <a href="/wiki/Logic" title="Logic">logic</a>, an axiom is a premise or starting point for reasoning.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <i>axiom</i> may be a "<a href="#Logical_axioms">logical axiom</a>" or a "<a href="#Non-logical_axioms">non-logical axiom</a>". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (<i>A</i> and <i>B</i>) implies <i>A</i>), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example <i>a</i> + 0 = <i>a</i> in integer arithmetic. </p><p>Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms".<sup id="cite_ref-properaxioms_5-0" class="reference"><a href="#cite_note-properaxioms-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain. </p><p>Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=1" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>axiom</i> comes from the <a href="/wiki/Greek_language" title="Greek language">Greek</a> word <span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἀξίωμα</span></span> (<i>axíōma</i>), a <a href="/wiki/Verbal_noun" title="Verbal noun">verbal noun</a> from the verb <span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἀξιόειν</span></span> (<i>axioein</i>), meaning "to deem worthy", but also "to require", which in turn comes from <span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἄξιος</span></span> (<i>áxios</i>), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the <a href="/wiki/Ancient_Greece" title="Ancient Greece">ancient Greek</a> <a href="/wiki/Philosopher" class="mw-redirect" title="Philosopher">philosophers</a> and <a href="/wiki/Greek_mathematics" title="Greek mathematics">mathematicians</a>, axioms were taken to be immediately evident propositions, foundational and common to many fields of investigation, and self-evidently true without any further argument or proof.<sup id="cite_ref-:0_7-0" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The root meaning of the word <i>postulate</i> is to "demand"; for instance, <a href="/wiki/Euclid" title="Euclid">Euclid</a> demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books, <a href="/wiki/Proclus" title="Proclus">Proclus</a> remarks that "<a href="/wiki/Geminus" title="Geminus">Geminus</a> held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property."<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Boethius" title="Boethius">Boethius</a> translated 'postulate' as <i>petitio</i> and called the axioms <i>notiones communes</i> but in later manuscripts this usage was not always strictly kept.<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 2023)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Historical_development">Historical development</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=2" title="Edit section: Historical development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Early_Greeks">Early Greeks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=3" title="Edit section: Early Greeks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (<a href="/wiki/Syllogisms" class="mw-redirect" title="Syllogisms">syllogisms</a>, <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a>) was developed by the ancient Greeks, and has become the core principle of modern mathematics. <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautologies</a> excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (<a href="/wiki/Theorem" title="Theorem">theorems</a>, in the case of mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms <i>axiom</i> and <i>postulate</i> hold a slightly different meaning for the present day mathematician, than they did for <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> and <a href="/wiki/Euclid" title="Euclid">Euclid</a>.<sup id="cite_ref-:0_7-1" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The ancient Greeks considered <a href="/wiki/Geometry" title="Geometry">geometry</a> as just one of several <a href="/wiki/Science" title="Science">sciences</a>, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's <a href="/wiki/Posterior_analytics" class="mw-redirect" title="Posterior analytics">posterior analytics</a> is a definitive exposition of the classical view.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>An "axiom", in classical terminology, referred to a <a href="/wiki/Self-evident" class="mw-redirect" title="Self-evident">self-evident</a> assumption common to many branches of science. A good example would be the assertion that: </p> <blockquote><p>When an equal amount is taken from equals, an equal amount results.</p></blockquote> <p>At the foundation of the various sciences lay certain additional <a href="/wiki/Hypothesis" title="Hypothesis">hypotheses</a> that were accepted without proof. Such a hypothesis was termed a <i>postulate</i>. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The classical approach is well-illustrated<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> by <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions). </p> <dl><dd><dl><dt>Postulates</dt></dl> <ol><li>It is possible to draw a <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a> from any point to any other point.</li> <li>It is possible to extend a <a href="/wiki/Line_segment" title="Line segment">line segment</a> continuously in both directions.</li> <li>It is possible to describe a <a href="/wiki/Circle" title="Circle">circle</a> with any center and any radius.</li> <li>It is true that all <a href="/wiki/Right_angle" title="Right angle">right angles</a> are equal to one another.</li> <li>("<a href="/wiki/Parallel_postulate" title="Parallel postulate">Parallel postulate</a>") It is true that, if a straight line falling on two straight lines make the <a href="/wiki/Polygon" title="Polygon">interior angles</a> on the same side less than two right angles, the two straight lines, if produced indefinitely, <a href="/wiki/Line-line_intersection" class="mw-redirect" title="Line-line intersection">intersect</a> on that side on which are the <a href="/wiki/Angle" title="Angle">angles</a> less than the two right angles.</li></ol></dd></dl> <dl><dd><dl><dt>Common notions</dt> <dd></dd></dl> <ol><li>Things which are equal to the same thing are also equal to one another.</li> <li>If equals are added to equals, the wholes are equal.</li> <li>If equals are subtracted from equals, the remainders are equal.</li> <li>Things which coincide with one another are equal to one another.</li> <li>The whole is greater than the part.</li></ol></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Modern_development">Modern development</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=4" title="Edit section: Modern development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositions</a>, theorems) and definitions. One must concede the need for <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notions</a>, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. <a href="/wiki/Alessandro_Padoa" title="Alessandro Padoa">Alessandro Padoa</a>, <a href="/wiki/Mario_Pieri" title="Mario Pieri">Mario Pieri</a>, and <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> were pioneers in this movement. </p><p>Structuralist mathematics goes further, and develops theories and axioms (e.g. <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group theory</a>, <a href="/wiki/Topological_space" title="Topological space">topology</a>, <a href="/wiki/Linear_space" class="mw-redirect" title="Linear space">vector spaces</a>) without <i>any</i> particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience. </p><p>When mathematicians employ the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all. </p><p>It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system. </p><p>Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and <a href="/wiki/Logicism" title="Logicism">mathematics itself can be regarded as a branch of logic</a>. <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a>, <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a>, <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, and <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a> are some of the key figures in this development. </p><p>Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions. </p><p>In the modern understanding, a set of axioms is any <a href="/wiki/Class_(set_theory)" title="Class (set theory)">collection</a> of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a>; it should be impossible to derive a contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom. </p><p>It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and the related demonstration of the consistency of those axioms. </p><p>In a wider context, there was an attempt to base all of mathematics on <a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor's</a> <a href="/wiki/Set_theory" title="Set theory">set theory</a>. Here, the emergence of <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a> and similar antinomies of <a href="/wiki/Na%C3%AFve_set_theory" class="mw-redirect" title="Naïve set theory">naïve set theory</a> raised the possibility that any such system could turn out to be inconsistent. </p><p>The formalist project suffered a setback a century ago, when <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel showed</a> that it is possible, for any sufficiently large set of axioms (<a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano's axioms</a>, for example) to construct a statement whose truth is independent of that set of axioms. As a <a href="/wiki/Corollary" title="Corollary">corollary</a>, Gödel proved that the consistency of a theory like <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> is an unprovable assertion within the scope of that theory.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, an <a href="/wiki/Infinite_set" title="Infinite set">infinite</a> but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern <a href="/wiki/Zermelo%E2%80%93Fraenkel_axioms" class="mw-redirect" title="Zermelo–Fraenkel axioms">Zermelo–Fraenkel axioms</a> for set theory. Furthermore, using techniques of <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a> (<a href="/wiki/Paul_Cohen" title="Paul Cohen">Cohen</a>) one can show that the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> (Cantor) is independent of the Zermelo–Fraenkel axioms.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Other_sciences">Other sciences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=5" title="Edit section: Other sciences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance, <a href="/wiki/Newton%27s_laws" class="mw-redirect" title="Newton's laws">Newton's laws</a> in classical mechanics, <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> in classical electromagnetism, <a href="/wiki/Einstein%27s_equation" class="mw-redirect" title="Einstein's equation">Einstein's equation</a> in general relativity, <a href="/wiki/Mendel%27s_laws" class="mw-redirect" title="Mendel's laws">Mendel's laws</a> of genetics, Darwin's <a href="/wiki/Natural_selection" title="Natural selection">Natural selection</a> law, etc. These founding assertions are usually called <i>principles</i> or <i>postulates</i> so as to distinguish from mathematical <i>axioms</i>. </p><p>As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying (<a href="/wiki/Falsifiability" title="Falsifiability">falsified</a>) the theory that the postulates install. A theory is considered valid as long as it has not been falsified. </p><p>Now, the transition between the mathematical axioms and scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidean geometry or differential calculus that they imply. It became more apparent when <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> first introduced <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> where the invariant quantity is no more the Euclidean length <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> (defined as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l^{2}=x^{2}+y^{2}+z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l^{2}=x^{2}+y^{2}+z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6ba8907e40d7d8e44b5fb03f64534517dbc3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.27ex; height:3.009ex;" alt="{\displaystyle l^{2}=x^{2}+y^{2}+z^{2}}"></span>) > but the Minkowski spacetime interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> (defined as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eded8efd4f0845c318941e89d4e2cd32e18bd91a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.463ex; height:3.009ex;" alt="{\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}}"></span>), and then <a href="/wiki/General_relativity" title="General relativity">general relativity</a> where flat Minkowskian geometry is replaced with <a href="/wiki/Pseudo-Riemannian" class="mw-redirect" title="Pseudo-Riemannian">pseudo-Riemannian</a> geometry on curved <a href="/wiki/Manifolds" class="mw-redirect" title="Manifolds">manifolds</a>. </p><p>In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The '<a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen school</a>' (<a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>, <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another '<a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">hidden variables</a>' approach was developed for some time by Albert Einstein, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="/wiki/David_Bohm" title="David Bohm">David Bohm</a>. It was created so as to try to give deterministic explanation to phenomena such as <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entanglement</a>. This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a> in 1935). Taking this idea seriously, <a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">John Bell</a> derived in 1964 a prediction that would lead to different experimental results (<a href="/wiki/Bell%27s_inequalities" class="mw-redirect" title="Bell's inequalities">Bell's inequalities</a>) in the Copenhagen and the Hidden variable case. The experiment was conducted first by <a href="/wiki/Alain_Aspect" title="Alain Aspect">Alain Aspect</a> in the early 1980s, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that the conceptual framework of quantum physics can be considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc.). </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_logic">Mathematical logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=6" title="Edit section: Mathematical logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the field of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, a clear distinction is made between two notions of axioms: <i>logical</i> and <i>non-logical</i> (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively). </p> <div class="mw-heading mw-heading3"><h3 id="Logical_axioms">Logical axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=7" title="Edit section: Logical axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These are certain <a href="/wiki/Formula_(mathematical_logic)" class="mw-redirect" title="Formula (mathematical logic)">formulas</a> in a <a href="/wiki/Formal_language" title="Formal language">formal language</a> that are <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">universally valid</a>, that is, formulas that are <a href="/wiki/Satisfiability" title="Satisfiability">satisfied</a> by every <a href="/wiki/Assignment_(mathematical_logic)" class="mw-redirect" title="Assignment (mathematical logic)">assignment</a> of values. Usually one takes as logical axioms <i>at least</i> some minimal set of tautologies that is sufficient for proving all <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a> in the language; in the case of <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a> more logical axioms than that are required, in order to prove <a href="/wiki/Logical_truth" title="Logical truth">logical truths</a> that are not tautologies in the strict sense. </p> <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=8" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading5"><h5 id="Propositional_logic">Propositional logic</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=9" title="Edit section: Propositional logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> it is common to take as logical axioms all formulae of the following forms, where <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, <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 \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"></span>, and <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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> can be any formulae of the language and where the included <a href="/wiki/Logical_connective" title="Logical connective">primitive connectives</a> are only "<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 \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span>" for <a href="/wiki/Negation" title="Negation">negation</a> of the immediately following proposition and "<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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span>" for <a href="/wiki/Entailment" class="mw-redirect" title="Entailment">implication</a> from antecedent to consequent propositions: </p> <ol><li><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 \phi \to (\psi \to \phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \to (\psi \to \phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30d0d40283ccbe848579fb56bec52a31798784c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.321ex; height:2.843ex;" alt="{\displaystyle \phi \to (\psi \to \phi )}"></span></li> <li><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 (\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451b60e906eb33473fd5c6129d2530cc93f00d98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.824ex; height:2.843ex;" alt="{\displaystyle (\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi ))}"></span></li> <li><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 (\lnot \phi \to \lnot \psi )\to (\psi \to \phi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lnot \phi \to \lnot \psi )\to (\psi \to \phi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3a4fd6f94a4008f83a52e46d05d5f2a70e56de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.006ex; height:2.843ex;" alt="{\displaystyle (\lnot \phi \to \lnot \psi )\to (\psi \to \phi ).}"></span></li></ol> <p>Each of these patterns is an <i><a href="/wiki/Axiom_schema" title="Axiom schema">axiom schema</a></i>, a rule for generating an infinite number of axioms. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, and <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> are <a href="/wiki/Propositional_variable" title="Propositional variable">propositional variables</a>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to (B\to A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to (B\to A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8692a5036a11936ef451eadfda6c4039a56249" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.288ex; height:2.843ex;" alt="{\displaystyle A\to (B\to A)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b41502a923fb8a5ab3a105586cd9e04a624712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.765ex; height:2.843ex;" alt="{\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))}"></span> are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and <i><a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a></i>, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with <i>modus ponens</i>. </p><p>Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>These axiom schemata are also used in the <a href="/wiki/Predicate_calculus" class="mw-redirect" title="Predicate calculus">predicate calculus</a>, but additional logical axioms are needed to include a quantifier in the calculus.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading5"><h5 id="First-order_logic">First-order logic</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=10" title="Edit section: First-order logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="border: 1px solid #CCCCCC; padding-left: 5px;"> <p><b>Axiom of Equality.</b><br />Let <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 {\mathfrak {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1cc2d02222bcba1e741979a145f0317df3cda81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.548ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {L}}}"></span> be a <a href="/wiki/First-order_language" class="mw-redirect" title="First-order language">first-order language</a>. For each variable <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, the below formula is universally valid. </p> <div class="center"> <p><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 x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d5b1d45c7aa8346eacace86bbb5cb0226f8fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:1.676ex;" alt="{\displaystyle x=x}"></span> </p> </div> </div> <p>This means that, for any <a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">variable symbol</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, the formula <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 x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d5b1d45c7aa8346eacace86bbb5cb0226f8fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:1.676ex;" alt="{\displaystyle x=x}"></span> can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d5b1d45c7aa8346eacace86bbb5cb0226f8fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:1.676ex;" alt="{\displaystyle x=x}"></span> (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that. </p><p>Another, more interesting example <a href="/wiki/Axiom_scheme" class="mw-redirect" title="Axiom scheme">axiom scheme</a>, is that which provides us with what is known as <b>Universal Instantiation</b>: </p> <div style="border: 1px solid #CCCCCC; padding-left: 5px;"> <p><b>Axiom scheme for Universal Instantiation.</b><br />Given a formula <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> in a first-order language <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 {\mathfrak {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1cc2d02222bcba1e741979a145f0317df3cda81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.548ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {L}}}"></span>, a variable <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and a <a href="/wiki/First_order_logic#Terms" class="mw-redirect" title="First order logic">term</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> that is <a href="/wiki/First-order_logic#Rules_of_inference" title="First-order logic">substitutable</a> 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, the below formula is universally valid. </p> <div class="center"> <p><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 \forall x\,\phi \to \phi _{t}^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>ϕ</mi> <mo stretchy="false">→</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\,\phi \to \phi _{t}^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a35f658409fe66bd83b7a3319c822b74648e1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.567ex; height:2.843ex;" alt="{\displaystyle \forall x\,\phi \to \phi _{t}^{x}}"></span> </p> </div> </div> <p>Where the symbol <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 \phi _{t}^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{t}^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c995755efbdd31d3545fb37ea81d1901b9c624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.558ex; height:2.843ex;" alt="{\displaystyle \phi _{t}^{x}}"></span> stands for the formula <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> with the term <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> substituted 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. (See <a href="/wiki/Substitution_of_variables" class="mw-redirect" title="Substitution of variables">Substitution of variables</a>.) In informal terms, this example allows us to state that, if we know that a certain property <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> holds for every <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and that <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> stands for a particular object in our structure, then we should be able to claim <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/941c1fb8073a146dc9938ef2aa43c2a84f6b2c97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.394ex; height:2.843ex;" alt="{\displaystyle P(t)}"></span>. Again, <i>we are claiming that the formula</i> <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 \forall x\phi \to \phi _{t}^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mi>ϕ</mi> <mo stretchy="false">→</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\phi \to \phi _{t}^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/818834dddd5dfebf016a0596076170170b519cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.18ex; height:2.843ex;" alt="{\displaystyle \forall x\phi \to \phi _{t}^{x}}"></span> <i>is valid</i>, that is, we must be able to give a "proof" of this fact, or more properly speaking, a <i>metaproof</i>. These examples are <i>metatheorems</i> of our theory of mathematical logic since we are dealing with the very concept of <i>proof</i> itself. Aside from this, we can also have <b>Existential Generalization</b>: </p> <div style="border: 1px solid #CCCCCC; padding-left: 5px;"> <p><b>Axiom scheme for Existential Generalization.</b> Given a formula <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> in a first-order language <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 {\mathfrak {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1cc2d02222bcba1e741979a145f0317df3cda81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.548ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {L}}}"></span>, a variable <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and a term <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> that is substitutable 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, the below formula is universally valid. </p> <div class="center"> <p><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 \phi _{t}^{x}\to \exists x\,\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mo stretchy="false">→</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{t}^{x}\to \exists x\,\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a5b41c0a3f75b2f4621f74ccae1dea79c20382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.567ex; height:2.843ex;" alt="{\displaystyle \phi _{t}^{x}\to \exists x\,\phi }"></span> </p> </div> </div> <div class="mw-heading mw-heading3"><h3 id="Non-logical_axioms">Non-logical axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=11" title="Edit section: Non-logical axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Non-logical axioms</b> are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> and the <a href="/wiki/Integer" title="Integer">integers</a>, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as <a href="/wiki/Group_(algebra)" class="mw-redirect" title="Group (algebra)">groups</a>). Thus non-logical axioms, unlike logical axioms, are not <i><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a></i>. Another name for a non-logical axiom is <i>postulate</i>.<sup id="cite_ref-properaxioms_5-1" class="reference"><a href="#cite_note-properaxioms-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Almost every modern <a href="/wiki/Mathematical_theory" class="mw-redirect" title="Mathematical theory">mathematical theory</a> starts from a given set of non-logical axioms, and it was thought that, in principle, every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.<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. (July 2011)">citation needed</span></a></i>]</sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="use of past tense without explanation of change (June 2019)">further explanation needed</span></a></i>]</sup> </p><p>Non-logical axioms are often simply referred to as <i>axioms</i> in mathematical <a href="/wiki/Discourse" title="Discourse">discourse</a>. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups. </p><p>Thus, an <i>axiom</i> is an elementary basis for a <a href="/wiki/Formal_system#Logical_system" title="Formal system">formal logic system</a> that together with the <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a> define a <a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive system</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Examples_2">Examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=12" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms. </p><p>Basic theories, such as <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a> and <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> with choice, abbreviated ZFC, or some very similar system of <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> like <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 set theory</a>, a <a href="/wiki/Conservative_extension" title="Conservative extension">conservative extension</a> of ZFC. Sometimes slightly stronger theories such as <a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley set theory</a> or set theory with a <a href="/wiki/Strongly_inaccessible_cardinal" class="mw-redirect" title="Strongly inaccessible cardinal">strongly inaccessible cardinal</a> allowing the use of a <a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck universe</a> is used, but in fact, most mathematicians can actually prove all they need in systems weaker than ZFC, such as <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>.<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 should include a citation (April 2016)">citation needed</span></a></i>]</sup> </p><p>The study of topology in mathematics extends all over through <a href="/wiki/Point_set_topology" class="mw-redirect" title="Point set topology">point set topology</a>, <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, and all the related paraphernalia, such as <a href="/wiki/Homology_theory" class="mw-redirect" title="Homology theory">homology theory</a>, <a href="/wiki/Homotopy_theory" title="Homotopy theory">homotopy theory</a>. The development of <i>abstract algebra</i> brought with itself <a href="/wiki/Group_theory" title="Group theory">group theory</a>, <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, and <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>. </p><p>This list could be expanded to include most fields of mathematics, including <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a>, <a href="/wiki/Probability" title="Probability">probability</a>, <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>, and <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. </p> <div class="mw-heading mw-heading5"><h5 id="Arithmetic">Arithmetic</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=13" title="Edit section: Arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a> are the most widely used <i>axiomatization</i> of <a href="/wiki/First-order_arithmetic" class="mw-redirect" title="First-order arithmetic">first-order arithmetic</a>. They are a set of axioms strong enough to prove many important facts about <a href="/wiki/Number_theory" title="Number theory">number theory</a> and they allowed Gödel to establish his famous <a href="/wiki/G%C3%B6del%27s_second_incompleteness_theorem" class="mw-redirect" title="Gödel's second incompleteness theorem">second incompleteness theorem</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>We have a language <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 {\mathfrak {L}}_{NT}=\{0,S\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}_{NT}=\{0,S\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65ad7e01097b5ef435cc35f4faf448f2dc26938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.515ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {L}}_{NT}=\{0,S\}}"></span> where <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> is a constant symbol and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is a <a href="/wiki/Unary_function" title="Unary function">unary function</a> and the following axioms: </p> <ol><li><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 \forall x.\lnot (Sx=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x.\lnot (Sx=0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be034a1e19bb9b81e7325147d36addc04a6e391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.106ex; height:2.843ex;" alt="{\displaystyle \forall x.\lnot (Sx=0)}"></span></li> <li><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 \forall x.\forall y.(Sx=Sy\to x=y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>.</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mi>x</mi> <mo>=</mo> <mi>S</mi> <mi>y</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x.\forall y.(Sx=Sy\to x=y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a762b3539275f23a1c0d75d3cc92f9f75dd953ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.727ex; height:2.843ex;" alt="{\displaystyle \forall x.\forall y.(Sx=Sy\to x=y)}"></span></li> <li><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 (\phi (0)\land \forall x.\,(\phi (x)\to \phi (Sx)))\to \forall x.\phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi (0)\land \forall x.\,(\phi (x)\to \phi (Sx)))\to \forall x.\phi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0cf4059d45003f4376de3bbd17022e8dce99e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.558ex; height:2.843ex;" alt="{\displaystyle (\phi (0)\land \forall x.\,(\phi (x)\to \phi (Sx)))\to \forall x.\phi (x)}"></span> for any <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 {\mathfrak {L}}_{NT}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {L}}_{NT}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c104e1572117f8ecf855f12f44f506f81c09495a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.396ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {L}}_{NT}}"></span> formula <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> with one free variable.</li></ol> <p>The standard structure is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}=\langle \mathbb {N} ,0,S\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">N</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}=\langle \mathbb {N} ,0,S\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94483778c18607d2936565770ff3a1e471968937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.249ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {N}}=\langle \mathbb {N} ,0,S\rangle }"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span> is the set of natural numbers, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is the <a href="/wiki/Successor_function" title="Successor function">successor function</a> and <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> is naturally interpreted as the number 0. </p> <div class="mw-heading mw-heading5"><h5 id="Euclidean_geometry">Euclidean geometry</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=14" title="Edit section: Euclidean geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Probably the oldest, and most famous, list of axioms are the 4 + 1 <a href="/wiki/Euclid%27s_postulates" class="mw-redirect" title="Euclid's postulates">Euclid's postulates</a> of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">plane geometry</a>. The axioms are referred to as "4 + 1" because for nearly two millennia the <a href="/wiki/Parallel_postulate" title="Parallel postulate">fifth (parallel) postulate</a> ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. One can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior <a href="/wiki/Angle" title="Angle">angles</a> of a <a href="/wiki/Triangle" title="Triangle">triangle</a> add up to exactly 180 degrees or less, respectively, and are known as Euclidean and <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic</a> geometries. If one also removes the second postulate ("a line can be extended indefinitely") then <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a> arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees. </p> <div class="mw-heading mw-heading5"><h5 id="Real_analysis">Real analysis</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=15" title="Edit section: Real analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The objectives of the study are within the domain of <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>. The real numbers are uniquely picked out (up to <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>) by the properties of a <i>Dedekind complete ordered field</i>, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires the use of <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a>. The <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorems</a> tell us that if we restrict ourselves to <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">non-standard analysis</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Role_in_mathematical_logic"><span id="role">Role in mathematical logic</span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=16" title="Edit section: Role in mathematical logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Deductive_systems_and_completeness">Deductive systems and completeness</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=17" title="Edit section: Deductive systems and completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b><a href="/wiki/Deductive" class="mw-redirect" title="Deductive">deductive</a> system</b> consists of a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> of logical axioms, a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> of non-logical axioms, and a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(\Gamma ,\phi )\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(\Gamma ,\phi )\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2389c9881d18d9e5de467d43228aea53616143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{(\Gamma ,\phi )\}}"></span> of <i>rules of inference</i>. A desirable property of a deductive system is that it be <b>complete</b>. A system is said to be complete if, for all formulas <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, </p> <div class="center"> <p><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 {\text{if }}\Sigma \models \phi {\text{ then }}\Sigma \vdash \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi mathvariant="normal">Σ</mi> <mo>⊨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> then </mtext> </mrow> <mi mathvariant="normal">Σ</mi> <mo>⊢</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{if }}\Sigma \models \phi {\text{ then }}\Sigma \vdash \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c57dcb3171a7d260ee7a400eace55910d44be9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.765ex; height:2.843ex;" alt="{\displaystyle {\text{if }}\Sigma \models \phi {\text{ then }}\Sigma \vdash \phi }"></span> </p> </div> <p>that is, for any statement that is a <i>logical consequence</i> of <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 \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> there actually exists a <i>deduction</i> of the statement from <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 \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span>. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a> establishes the completeness of a certain commonly used type of deductive system. </p><p>Note that "completeness" has a different meaning here than it does in the context of <a href="/wiki/G%C3%B6del%27s_first_incompleteness_theorem" class="mw-redirect" title="Gödel's first incompleteness theorem">Gödel's first incompleteness theorem</a>, which states that no <i>recursive</i>, <i>consistent</i> set of non-logical axioms <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 \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> of the Theory of Arithmetic is <i>complete</i>, in the sense that there will always exist an arithmetic statement <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> such that neither <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> nor <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 \lnot \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbee468759b489b9639ad4cd30aa566ac89b2096" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.936ex; height:2.509ex;" alt="{\displaystyle \lnot \phi }"></span> can be proved from the given set of axioms. </p><p>There is thus, on the one hand, the notion of <i>completeness of a deductive system</i> and on the other hand that of <i>completeness of a set of non-logical axioms</i>. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another. </p> <div class="mw-heading mw-heading3"><h3 id="Further_discussion">Further discussion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=18" title="Edit section: Further discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Early <a href="/wiki/Mathematician" title="Mathematician">mathematicians</a> regarded <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">axiomatic geometry</a> as a model of <a href="/wiki/Physical_space" class="mw-redirect" title="Physical space">physical space</a>, and obviously, there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as <a href="/wiki/Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a> made elaborate efforts to derive them from traditional arithmetic. <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Galois</a> showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, and <a href="/wiki/Abstract_algebra" title="Abstract algebra">modern algebra</a> was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.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 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.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-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Although not complete; some of the stated results did not actually follow from the stated postulates and common notions.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Hilbert also made explicit the assumptions that Euclid used in his proofs but did not list in his common notions and postulates.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiom&action=edit&section=21" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Cf. axiom, n., etymology. <i>Oxford English Dictionary</i>, accessed 2012-04-28.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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="CITEREFStevensonLindberg2015" class="citation book cs1">Stevenson, Angus; Lindberg, Christine A., eds. (2015). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.oxfordreference.com/display/10.1093/acref/9780195392883.001.0001/m_en_us1224100"><i>New Oxford American Dictionary</i></a></span> (3rd ed.). Oxford University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Facref%2F9780195392883.001.0001">10.1093/acref/9780195392883.001.0001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199891535" title="Special:BookSources/9780199891535"><bdi>9780199891535</bdi></a>. <q>a statement or proposition that is regarded as being established, accepted, or self-evidently true</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=New+Oxford+American+Dictionary&rft.edition=3rd&rft.pub=Oxford+University+Press&rft.date=2015&rft_id=info%3Adoi%2F10.1093%2Facref%2F9780195392883.001.0001&rft.isbn=9780199891535&rft_id=https%3A%2F%2Fwww.oxfordreference.com%2Fdisplay%2F10.1093%2Facref%2F9780195392883.001.0001%2Fm_en_us1224100&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">"A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. <i>Oxford English Dictionary</i> Online, accessed 2012-04-28. Cf. Aristotle, <i><a href="/wiki/Posterior_Analytics" title="Posterior Analytics">Posterior Analytics</a></i> I.2.72a18-b4.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">"A proposition (whether true or false)" axiom, n., definition 2. <i>Oxford English Dictionary</i> Online, accessed 2012-04-28.</span> </li> <li id="cite_note-properaxioms-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-properaxioms_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-properaxioms_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Mendelson, "3. First-Order Theories: Proper Axioms" of Ch. 2</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">See for example <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaddy1988" class="citation journal cs1">Maddy, Penelope (June 1988). "Believing the Axioms, I". <i>Journal of Symbolic Logic</i>. <b>53</b> (2): 481–511. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2274520">10.2307/2274520</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2274520">2274520</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Believing+the+Axioms%2C+I&rft.volume=53&rft.issue=2&rft.pages=481-511&rft.date=1988-06&rft_id=info%3Adoi%2F10.2307%2F2274520&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2274520%23id-name%3DJSTOR&rft.aulast=Maddy&rft.aufirst=Penelope&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span> for a <a href="/wiki/Mathematical_realism" class="mw-redirect" title="Mathematical realism">realist</a> view.</span> </li> <li id="cite_note-:0-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.ptta.pl/pef/haslaen/a/axiom.pdf">"Axiom — Powszechna Encyklopedia Filozofii"</a> <span class="cs1-format">(PDF)</span>. <i>Polskie Towarzystwo Tomasza z Akwinu</i>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.ptta.pl/pef/haslaen/a/axiom.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 9 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Polskie+Towarzystwo+Tomasza+z+Akwinu&rft.atitle=Axiom+%E2%80%94+Powszechna+Encyklopedia+Filozofii&rft_id=http%3A%2F%2Fwww.ptta.pl%2Fpef%2Fhaslaen%2Fa%2Faxiom.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span></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">Wolff, P. <i>Breakthroughs in Mathematics</i>, 1963, New York: New American Library, pp 47–48</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1956" class="citation book cs1"><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, T. L.</a> (1956). <i>The Thirteen Books of Euclid's Elements</i>. New York: Dover. p. 200.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.place=New+York&rft.pages=200&rft.pub=Dover&rft.date=1956&rft.aulast=Heath&rft.aufirst=T.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/biography/Aristotle">"Aristotle | Biography, Works, Quotes, Philosophy, Ethics, & Facts | Britannica"</a>. <i>www.britannica.com</i>. 8 October 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">14 November</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.britannica.com&rft.atitle=Aristotle+%7C+Biography%2C+Works%2C+Quotes%2C+Philosophy%2C+Ethics%2C+%26+Facts+%7C+Britannica&rft.date=2024-10-08&rft_id=https%3A%2F%2Fwww.britannica.com%2Fbiography%2FAristotle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Aristotle, Metaphysics Bk IV, Chapter 3, 1005b "Physics also is a kind of Wisdom, but it is not the first kind. – And the attempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic; for they should know these things already when they come to a special study, and not be inquiring into them while they are listening to lectures on it." W.D. Ross translation, in The Basic Works of Aristotle, ed. Richard McKeon, (Random House, New York, 1941)</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">For more, see <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaatikainen2018" class="citation cs2">Raatikainen, Panu (2018), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/">"Gödel's Incompleteness Theorems"</a>, in Zalta, Edward N. (ed.), <i>The Stanford Encyclopedia of Philosophy</i> (Fall 2018 ed.), Metaphysics Research Lab, Stanford University<span class="reference-accessdate">, retrieved <span class="nowrap">19 October</span> 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=G%C3%B6del%27s+Incompleteness+Theorems&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Fall+2018&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2018&rft.aulast=Raatikainen&rft.aufirst=Panu&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Ffall2018%2Fentries%2Fgoedel-incompleteness%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoellner2019" class="citation cs2">Koellner, Peter (2019), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/spr2019/entries/continuum-hypothesis/">"The Continuum Hypothesis"</a>, in Zalta, Edward N. (ed.), <i>The Stanford Encyclopedia of Philosophy</i> (Spring 2019 ed.), Metaphysics Research Lab, Stanford University<span class="reference-accessdate">, retrieved <span class="nowrap">19 October</span> 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Continuum+Hypothesis&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Spring+2019&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2019&rft.aulast=Koellner&rft.aufirst=Peter&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fspr2019%2Fentries%2Fcontinuum-hypothesis%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Mendelson, "6. Other Axiomatizations" of Ch. 1</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Mendelson, "3. First-Order Theories" of Ch. 2</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Mendelson, "5. The Fixed Point Theorem. Gödel's Incompleteness Theorem" of Ch. 2</span> </li> </ol></div></div> <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=Axiom&action=edit&section=22" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Mendelson, Elliot (1987). <i>Introduction to mathematical logic.</i> Belmont, California: Wadsworth & Brooks. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-06624-0" title="Special:BookSources/0-534-06624-0">0-534-06624-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Cook_Wilson1889" class="citation cs2"><a href="/wiki/John_Cook_Wilson" title="John Cook Wilson">John Cook Wilson</a> (1889), <i><a href="https://en.wikisource.org/wiki/On_an_Evolutionist_Theory_of_Axioms" class="extiw" title="s:On an Evolutionist Theory of Axioms">On an Evolutionist Theory of Axioms: inaugural lecture delivered October 15, 1889</a></i> (1st ed.), Oxford, <a href="/wiki/WDQ_(identifier)" class="mw-redirect" title="WDQ (identifier)">Wikidata</a> <a href="https://www.wikidata.org/wiki/Q26720682" class="extiw" title="d:Q26720682">Q26720682</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+an+Evolutionist+Theory+of+Axioms%3A+inaugural+lecture+delivered+October+15%2C+1889&rft.place=Oxford&rft.edition=1st&rft.date=1889&rft.au=John+Cook+Wilson&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAxiom" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></li></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=Axiom&action=edit&section=23" 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 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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"><span><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/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/axiom" class="extiw" title="wiktionary:axiom">axiom</a></b></i> or <i><b><a href="https://en.wiktionary.org/wiki/given" class="extiw" title="wiktionary:given">given</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has the text of the <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">1911 <i>Encyclopædia Britannica</i></a> article "<span style="font-weight:bold;"><a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Axiom" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Axiom">Axiom</a></span>".</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://philpapers.org/s/axiom">Axiom</a> at <a href="/wiki/PhilPapers" title="PhilPapers">PhilPapers</a></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/Axiom">Axiom</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li> <li><a rel="nofollow" class="external text" href="http://us.metamath.org/mpegif/mmset.html#axioms"><i>Metamath</i> axioms page</a></li></ul> <div class="navbox-styles"><style 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete 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