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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Ordinals_extend_the_natural_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ordinals_extend_the_natural_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Ordinals extend the natural numbers</span> </div> </a> <ul id="toc-Ordinals_extend_the_natural_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definitions</span> </div> </a> <button aria-controls="toc-Definitions-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 Definitions subsection</span> </button> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-Well-ordered_sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Well-ordered_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Well-ordered sets</span> </div> </a> <ul id="toc-Well-ordered_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_of_an_ordinal_as_an_equivalence_class" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_of_an_ordinal_as_an_equivalence_class"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Definition of an ordinal as an equivalence class</span> </div> </a> <ul id="toc-Definition_of_an_ordinal_as_an_equivalence_class-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Von_Neumann_definition_of_ordinals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Von_Neumann_definition_of_ordinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Von Neumann definition of ordinals</span> </div> </a> <ul id="toc-Von_Neumann_definition_of_ordinals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Other definitions</span> </div> </a> <ul id="toc-Other_definitions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transfinite_sequence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transfinite_sequence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Transfinite sequence</span> </div> </a> <ul id="toc-Transfinite_sequence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transfinite_induction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transfinite_induction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Transfinite induction</span> </div> </a> <button aria-controls="toc-Transfinite_induction-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 Transfinite induction subsection</span> </button> <ul id="toc-Transfinite_induction-sublist" class="vector-toc-list"> <li id="toc-Transfinite_recursion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transfinite_recursion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Transfinite recursion</span> </div> </a> <ul id="toc-Transfinite_recursion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Successor_and_limit_ordinals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Successor_and_limit_ordinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Successor and limit ordinals</span> </div> </a> <ul id="toc-Successor_and_limit_ordinals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Indexing_classes_of_ordinals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Indexing_classes_of_ordinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Indexing classes of ordinals</span> </div> </a> <ul id="toc-Indexing_classes_of_ordinals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_unbounded_sets_and_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_unbounded_sets_and_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Closed unbounded sets and classes</span> </div> </a> <ul id="toc-Closed_unbounded_sets_and_classes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Arithmetic_of_ordinals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arithmetic_of_ordinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Arithmetic of ordinals</span> </div> </a> <ul id="toc-Arithmetic_of_ordinals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordinals_and_cardinals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ordinals_and_cardinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Ordinals and cardinals</span> </div> </a> <button aria-controls="toc-Ordinals_and_cardinals-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 Ordinals and cardinals subsection</span> </button> <ul id="toc-Ordinals_and_cardinals-sublist" class="vector-toc-list"> <li id="toc-Initial_ordinal_of_a_cardinal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Initial_ordinal_of_a_cardinal"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Initial ordinal of a cardinal</span> </div> </a> <ul id="toc-Initial_ordinal_of_a_cardinal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cofinality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cofinality"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Cofinality</span> </div> </a> <ul id="toc-Cofinality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Some_"large"_countable_ordinals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Some_"large"_countable_ordinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Some "large" countable ordinals</span> </div> </a> <ul id="toc-Some_"large"_countable_ordinals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology_and_ordinals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topology_and_ordinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Topology and ordinals</span> </div> </a> <ul id="toc-Topology_and_ordinals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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">Ordinal number</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 41 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-41" 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">41 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AA%D8%B1%D8%AA%D9%8A%D8%A8%D9%8A" title="عدد ترتيبي – Arabic" lang="ar" hreflang="ar" data-title="عدد ترتيبي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/S%C4%B1ra_say%C4%B1" title="Sıra sayı – Azerbaijani" lang="az" hreflang="az" data-title="Sıra sayı" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_ordinal" title="Nombre ordinal – Catalan" lang="ca" hreflang="ca" data-title="Nombre ordinal" 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%99%C4%95%D1%80%D0%BA%D0%B5_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%C4%95" title="Йĕрке хисепĕ – Chuvash" lang="cv" hreflang="cv" data-title="Йĕрке хисепĕ" 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/Ordin%C3%A1ln%C3%AD_%C4%8D%C3%ADslo" title="Ordinální číslo – Czech" lang="cs" hreflang="cs" data-title="Ordinální číslo" 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/Trefnolyn" title="Trefnolyn – Welsh" lang="cy" hreflang="cy" data-title="Trefnolyn" 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/Ordinaltal" title="Ordinaltal – Danish" lang="da" hreflang="da" data-title="Ordinaltal" 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/Ordinalzahl" title="Ordinalzahl – German" lang="de" hreflang="de" data-title="Ordinalzahl" 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/Ordinaalarv" title="Ordinaalarv – Estonian" lang="et" hreflang="et" data-title="Ordinaalarv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_ordinal_(teor%C3%ADa_de_conjuntos)" title="Número ordinal (teoría de conjuntos) – Spanish" lang="es" hreflang="es" data-title="Número ordinal (teoría de conjuntos)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ordonombro" title="Ordonombro – Esperanto" lang="eo" hreflang="eo" data-title="Ordonombro" 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/Zenbaki_ordinal" title="Zenbaki ordinal – Basque" lang="eu" hreflang="eu" data-title="Zenbaki ordinal" 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%B9%D8%AF%D8%AF_%D8%AA%D8%B1%D8%AA%DB%8C%D8%A8%DB%8C" title="عدد ترتیبی – Persian" lang="fa" hreflang="fa" data-title="عدد ترتیبی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_ordinal" title="Nombre ordinal – French" lang="fr" hreflang="fr" data-title="Nombre ordinal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_ordinal" title="Número ordinal – Galician" lang="gl" hreflang="gl" data-title="Número ordinal" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%9C%EC%84%9C%EC%88%98" title="순서수 – Korean" lang="ko" hreflang="ko" data-title="순서수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Ordinala_nombro" title="Ordinala nombro – Ido" lang="io" hreflang="io" data-title="Ordinala nombro" 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/Bilangan_ordinal" title="Bilangan ordinal – Indonesian" lang="id" hreflang="id" data-title="Bilangan ordinal" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ra%C3%B0tala" title="Raðtala – Icelandic" lang="is" hreflang="is" data-title="Raðtala" 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/Numero_ordinale_(teoria_degli_insiemi)" title="Numero ordinale (teoria degli insiemi) – Italian" lang="it" hreflang="it" data-title="Numero ordinale (teoria degli insiemi)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A1%D7%95%D7%93%D7%A8" title="מספר סודר – Hebrew" lang="he" hreflang="he" data-title="מספר סודר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9E%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D0%BB_%D1%81%D0%B0%D0%BD" title="Ординал сан – Kazakh" lang="kk" hreflang="kk" data-title="Ординал сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_ordinal" title="Numer ordinal – Lombard" lang="lmo" hreflang="lmo" data-title="Numer ordinal" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rendsz%C3%A1m_(halmazelm%C3%A9let)" title="Rendszám (halmazelmélet) – Hungarian" lang="hu" hreflang="hu" data-title="Rendszám (halmazelmélet)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B4%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реден број – Macedonian" lang="mk" hreflang="mk" data-title="Реден број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_ordinal" title="Nombor ordinal – Malay" lang="ms" hreflang="ms" data-title="Nombor ordinal" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ordinaalgetal" title="Ordinaalgetal – Dutch" lang="nl" hreflang="nl" data-title="Ordinaalgetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%A0%86%E5%BA%8F%E6%95%B0" title="順序数 – Japanese" lang="ja" hreflang="ja" data-title="順序数" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_porz%C4%85dkowe" title="Liczby porządkowe – Polish" lang="pl" hreflang="pl" data-title="Liczby porządkowe" 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/N%C3%BAmero_ordinal" title="Número ordinal – Portuguese" lang="pt" hreflang="pt" data-title="Número ordinal" 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/Num%C4%83r_ordinal" title="Număr ordinal – Romanian" lang="ro" hreflang="ro" data-title="Număr ordinal" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D1%8F%D0%B4%D0%BA%D0%BE%D0%B2%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Порядковое число – Russian" lang="ru" hreflang="ru" data-title="Порядковое число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ordinalno_%C5%A1tevilo" title="Ordinalno število – Slovenian" lang="sl" hreflang="sl" data-title="Ordinalno število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ordinaali" title="Ordinaali – Finnish" lang="fi" hreflang="fi" data-title="Ordinaali" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ordinaltal" title="Ordinaltal – Swedish" lang="sv" hreflang="sv" data-title="Ordinaltal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Ordinal_numbers&redirect=no" class="mw-redirect" title="Ordinal numbers">Ordinal numbers</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Generalization of "n-th" to infinite cases</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 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For number words denoting a position in a sequence ("first", "second", "third", etc.), see <a href="/wiki/Ordinal_numeral" title="Ordinal numeral">Ordinal numeral</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output 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One turn of the spiral corresponds to the mapping <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\alpha )=\omega (1+\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\alpha )=\omega (1+\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a08acf61c94f0a4671f672f33673a6da87b11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.42ex; height:2.843ex;" alt="{\displaystyle f(\alpha )=\omega (1+\alpha )}"></span>⁠</span>. Since <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> has <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 \omega ^{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa5636cabcea62c44c8b91fd9095e06054a6fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.7ex; height:2.343ex;" alt="{\displaystyle \omega ^{\omega }}"></span> as least fixed point, larger ordinal numbers cannot be represented on this diagram.</figcaption></figure> <p>In <a href="/wiki/Set_theory" title="Set theory">set theory</a>, an <b>ordinal number</b>, or <b>ordinal</b>, is a generalization of <a href="/wiki/Ordinal_numeral" title="Ordinal numeral">ordinal numerals</a> (first, second, <span class="texhtml mvar" style="font-style:italic;">n</span>th, etc.) aimed to extend <a href="/wiki/Enumeration" title="Enumeration">enumeration</a> to <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a>.<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> </p><p>A finite set can be enumerated by successively labeling each element with the least <a href="/wiki/Natural_number" title="Natural number">natural number</a> that has not been previously used. To extend this process to various <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a>, ordinal numbers are defined more generally using <a href="/wiki/Linearly_ordered" class="mw-redirect" title="Linearly ordered">linearly ordered</a> <a href="/wiki/Greek_letters" class="mw-redirect" title="Greek letters">greek letter</a> <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> that include the natural numbers and have the property that every set of ordinals has a <a href="/wiki/Greatest_element_and_least_element" title="Greatest element and least element">least or "smallest" element</a> (this is needed for giving a meaning to "the least unused element").<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> This more general definition allows us to define an ordinal number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> (omega) to be the least element that is greater than every natural number, along with ordinal numbers <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64eefa705e5e12f34fc2fe138f4897cd4980f374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.449ex; height:2.343ex;" alt="{\displaystyle \omega +1}"></span>⁠</span>, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega +2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega +2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3ab03d7453dd4987880c536b907b7c48303f7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.449ex; height:2.343ex;" alt="{\displaystyle \omega +2}"></span>⁠</span>, etc., which are even greater than <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>⁠</span>. </p><p>A linear order such that every non-empty subset has a least element is called a <a href="/wiki/Well-order" title="Well-order">well-order</a>. The <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> implies that every set can be well-ordered, and given two well-ordered sets, one is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to an <a href="/wiki/Initial_segment" class="mw-redirect" title="Initial segment">initial segment</a> of the other. So ordinal numbers exist and are essentially unique. </p><p>Ordinal numbers are distinct from <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be <a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">added, multiplied, and exponentiated</a>, although none of these operations are <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>. </p><p>Ordinals were introduced by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> in 1883<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 order to accommodate infinite sequences and classify <a href="/wiki/Derived_set_(mathematics)" title="Derived set (mathematics)">derived sets</a>, which he had previously introduced in 1872 while studying the uniqueness of <a href="/wiki/Trigonometric_series" title="Trigonometric series">trigonometric series</a>.<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ordinals_extend_the_natural_numbers">Ordinals extend the natural numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=1" title="Edit section: Ordinals extend the natural numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="omega"></span> A <a href="/wiki/Natural_number" title="Natural number">natural number</a> (which, in this context, includes the number <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a>) can be used for two purposes: to describe the <i>size</i> of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, or to describe the <i>position</i> of an element in a sequence. When restricted to finite sets, these two concepts coincide, since all <a href="/wiki/Linear_order" class="mw-redirect" title="Linear order">linear orders</a> of a finite set are <a href="/wiki/Order_isomorphism" title="Order isomorphism">isomorphic</a>. </p><p>When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>), there are many nonisomorphic <a href="/wiki/Well-ordering" class="mw-redirect" title="Well-ordering">well-orderings</a> of any infinite set, as explained below. </p><p>Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called <a href="/wiki/Well-order" title="Well-order">well-ordered</a>. A well-ordered set is a <a href="/wiki/Totally_ordered" class="mw-redirect" title="Totally ordered">totally ordered</a> set (an <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">ordered</a> set such that, given two distinct elements, one is less than the other) in which every non-empty subset has a least element. Equivalently, assuming the <a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">axiom of dependent choice</a>, it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the <i>order type</i> of the set. </p><p>Any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinals <i>identifies</i> each ordinal <i>as</i> the set of ordinals that precede it. For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set <i>S</i> of ordinals that is <a href="/wiki/Downward_closed" class="mw-redirect" title="Downward closed">downward closed</a> — meaning that for any ordinal α in <i>S</i> and any ordinal β < α, β is also in <i>S</i> — is (or can be identified with) an ordinal. </p><p>This definition of ordinals in terms of sets allows for infinite ordinals. The smallest infinite ordinal is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>⁠</span>, which can be identified with the set of natural numbers (so that the ordinal associated with every natural number precedes <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>). Indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed, it can be identified with the ordinal associated with it. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ordinal_ww.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Ordinal_ww.svg/256px-Ordinal_ww.svg.png" decoding="async" width="256" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Ordinal_ww.svg/384px-Ordinal_ww.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Ordinal_ww.svg/512px-Ordinal_ww.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>A graphical "matchstick" representation of the ordinal ω<sup>2</sup>. Each stick corresponds to an ordinal of the form ω·<i>m</i>+<i>n</i> where <i>m</i> and <i>n</i> are natural numbers.</figcaption></figure> <p>Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, ... After <i>all</i> natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω·<i>m</i>+<i>n</i>, where <i>m</i> and <i>n</i> are natural numbers) must itself have an ordinal associated with it: and that is ω<sup>2</sup>. Further on, there will be ω<sup>3</sup>, then ω<sup>4</sup>, and so on, and ω<sup>ω</sup>, then ω<sup>ω<sup>ω</sup></sup>, then later ω<sup>ω<sup>ω<sup>ω</sup></sup></sup>, and even later ε<sub>0</sub> (<a href="/wiki/Epsilon_number" title="Epsilon number">epsilon nought</a>) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable</a> ordinal is the set of all countable ordinals, expressed as <a href="/wiki/First_uncountable_ordinal" title="First uncountable ordinal">ω<sub>1</sub></a> or <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 \Omega }"></span>⁠</span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=2" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Well-ordered_sets">Well-ordered sets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=3" title="Edit section: Well-ordered sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Well-order" title="Well-order">well-ordered</a> set, every non-empty subset contains a distinct smallest element. Given the <a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">axiom of dependent choice</a>, this is equivalent to saying that the set is <a href="/wiki/Totally_ordered" class="mw-redirect" title="Totally ordered">totally ordered</a> and there is no infinite decreasing sequence (the latter being easier to visualize). In practice, the importance of well-ordering is justified by the possibility of applying <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a>, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by a "lower" step—then the computation will terminate. </p><p>It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an <a href="/wiki/Order_isomorphism" title="Order isomorphism">order isomorphism</a>, and the two well-ordered sets are said to be order-isomorphic or <i>similar</i> (with the understanding that this is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>). </p><p>Formally, if a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> ≤ is defined on the set <i>S</i>, and a partial order ≤' is defined on the set <i>S' </i>, then the <a href="/wiki/Partially_ordered_set" title="Partially ordered set">posets</a> (<i>S</i>,≤) and (<i>S' </i>,≤') are <a href="/wiki/Order_isomorphic" class="mw-redirect" title="Order isomorphic">order isomorphic</a> if there is a <a href="/wiki/Bijection" title="Bijection">bijection</a> <i>f</i> that preserves the ordering. That is, <i>f</i>(<i>a</i>) ≤' <i>f</i>(<i>b</i>) if and only if <i>a</i> ≤ <i>b</i>. Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a <a href="/wiki/Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">"canonical" representative</a> of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Every <i>well-ordered</i> set (<i>S</i>,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the <i>order type</i> of (<i>S</i>,<). </p><p>Essentially, an ordinal is intended to be defined as an <a href="/wiki/Isomorphism_class" class="mw-redirect" title="Isomorphism class">isomorphism class</a> of well-ordered sets: that is, as an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> for the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the <i><a href="/wiki/Order_type" title="Order type">order type</a></i> of any set in the class. </p> <div class="mw-heading mw-heading3"><h3 id="Definition_of_an_ordinal_as_an_equivalence_class">Definition of an ordinal as an equivalence class</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=4" title="Edit section: Definition of an ordinal as an equivalence class"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The original definition of ordinal numbers, found for example in the <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i>, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZF</a> and related systems of <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> because these equivalence classes are too large to form a set. However, this definition still can be used in <a href="/wiki/Type_theory" title="Type theory">type theory</a> and in Quine's axiomatic set theory <a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a> and related systems (where it affords a rather surprising alternative solution to the <a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a> of the largest ordinal). </p> <div class="mw-heading mw-heading3"><h3 id="Von_Neumann_definition_of_ordinals">Von Neumann definition of ordinals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=5" title="Edit section: Von Neumann definition of ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Set-theoretic_definition_of_natural_numbers" title="Set-theoretic definition of natural numbers">Set-theoretic definition of natural numbers</a> and <a href="/wiki/Zermelo_ordinals" class="mw-redirect" title="Zermelo ordinals">Zermelo ordinals</a></div> <table class="floatright" style="background-color: #f8f9fa; border: 1px solid #a2a9b1; margin: 0.5em 0 0.5em 1em; padding: 0.2em; color:black;"> <caption>First several von Neumann ordinals </caption> <tbody><tr> <th scope="row">0 </th> <td>=</td> <td>{} </td> <td>=</td> <td>∅ </td></tr> <tr> <th scope="row">1 </th> <td>=</td> <td>{0} </td> <td>=</td> <td>{∅} </td></tr> <tr> <th scope="row">2 </th> <td>=</td> <td>{0,1} </td> <td>=</td> <td>{∅,{∅}} </td></tr> <tr> <th scope="row">3 </th> <td>=</td> <td>{0,1,2} </td> <td>=</td> <td>{∅,{∅},{∅,{∅}}} </td></tr> <tr> <th scope="row">4 </th> <td>=</td> <td>{0,1,2,3} </td> <td>=</td> <td>{∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}} </td></tr></tbody></table> <p>Rather than defining an ordinal as an <i>equivalence class</i> of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number. </p><p>For each well-ordered set <span class="texhtml mvar" style="font-style:italic;">T</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 a\mapsto T_{<a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto T_{<a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd6610b401dec26ad721b502f76953f65f98d061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.582ex; height:2.509ex;" alt="{\displaystyle a\mapsto T_{<a}}"></span> defines an <a href="/wiki/Order_isomorphism" title="Order isomorphism">order isomorphism</a> between <span class="texhtml mvar" style="font-style:italic;">T</span> and the set of all subsets of <span class="texhtml mvar" style="font-style:italic;">T</span> having the form <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_{<a}:=\{x\in T\mid x<a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mi>a</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>T</mi> <mo>∣</mo> <mi>x</mi> <mo><</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{<a}:=\{x\in T\mid x<a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70314818e6d2834b4b8c145d169c03feef56e5ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.21ex; height:2.843ex;" alt="{\displaystyle T_{<a}:=\{x\in T\mid x<a\}}"></span> ordered by inclusion. This motivates the standard definition, suggested by <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> at the age of 19, now called definition of <b>von Neumann ordinals</b>: "each ordinal is the well-ordered set of all smaller ordinals". In symbols, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =[0,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =[0,\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb1e33db4184506b820c5fd9eb925a8a1b2840e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.557ex; height:2.843ex;" alt="{\displaystyle \lambda =[0,\lambda )}"></span>⁠</span>.<sup id="cite_ref-von_Neumann1923pp199-208_6-0" class="reference"><a href="#cite_note-von_Neumann1923pp199-208-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Formally: </p> <dl><dd>A set <i>S</i> is an ordinal <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>S</i> is <a href="/wiki/Strict_order" class="mw-redirect" title="Strict order">strictly</a> well-ordered with respect to set membership and every element of <i>S</i> is also a subset of <i>S</i>.</dd></dl> <p>The natural numbers are thus ordinals by this definition. For instance, 2 is an element of <span class="texhtml">4 = {0, 1, 2, 3},</span> and 2 is equal to <span class="texhtml">{0, 1}</span> and so it is a subset of <span class="texhtml">{0, 1, 2, 3}</span>. </p><p>It can be shown by <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a> that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective function</a> between them. </p><p>Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals <i>S</i> and <i>T</i>, <i>S</i> is an element of <i>T</i> if and only if <i>S</i> is a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of <i>T</i>. Moreover, either <i>S</i> is an element of <i>T</i>, or <i>T</i> is an element of <i>S</i>, or they are equal. So every set of ordinals is <a href="/wiki/Total_order" title="Total order">totally ordered</a>. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered. </p><p>Consequently, every ordinal <i>S</i> is a set having as elements precisely the ordinals smaller than <i>S</i>. For example, every set of ordinals has a <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the <a href="/wiki/Axiom_of_union" title="Axiom of union">axiom of union</a>. </p><p>The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its <i>strict</i> ordering by membership. This is the <a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a>. The class of all ordinals is variously called "Ord", "ON", or "∞". </p><p>An ordinal is <a href="/wiki/Finite_set" title="Finite set">finite</a> if and only if the opposite order is also well-ordered, which is the case if and only if each of its non-empty subsets has a <a href="/wiki/Greatest_element_and_least_element" title="Greatest element and least element">greatest element</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_definitions">Other definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=6" title="Edit section: Other definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are other modern formulations of the definition of ordinal. For example, assuming the <a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">axiom of regularity</a>, the following are equivalent for a set <i>x</i>: </p> <ul><li><i>x</i> is a (von Neumann) ordinal,</li> <li><i>x</i> is a <a href="/wiki/Transitive_set" title="Transitive set">transitive set</a>, and set membership is <a href="/wiki/Trichotomy_(mathematics)" class="mw-redirect" title="Trichotomy (mathematics)">trichotomous</a> on <i>x</i>,</li> <li><i>x</i> is a transitive set <a href="/wiki/Total_order" title="Total order">totally ordered</a> by set inclusion,</li> <li><i>x</i> is a transitive set of transitive sets.</li></ul> <p>These definitions cannot be used in <a href="/wiki/Non-well-founded_set_theory" title="Non-well-founded set theory">non-well-founded set theories</a>. In set theories with <a href="/wiki/Urelement" title="Urelement">urelements</a>, one has to further make sure that the definition excludes urelements from appearing in ordinals. </p> <div class="mw-heading mw-heading2"><h2 id="Transfinite_sequence">Transfinite sequence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=7" title="Edit section: Transfinite sequence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If α is any ordinal and <i>X</i> is a set, an α-indexed sequence of elements of <i>X</i> is a function from α to <i>X</i>. This concept, a <b>transfinite sequence</b> (if α is infinite) or <i>ordinal-indexed sequence</i>, is a generalization of the concept of a <a href="/wiki/Sequence" title="Sequence">sequence</a>. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a <a href="/wiki/Tuple_(mathematics)" class="mw-redirect" title="Tuple (mathematics)">tuple</a>, a.k.a. <a href="/wiki/String_(computer_science)" title="String (computer science)">string</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Transfinite_induction">Transfinite induction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=8" title="Edit section: Transfinite induction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></div> <p>Transfinite induction holds in any <a href="/wiki/Well-order" title="Well-order">well-ordered</a> set, but it is so important in relation to ordinals that it is worth restating here. </p> <dl><dd>Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals.</dd></dl> <p>That is, if <i>P</i>(α) is true whenever <i>P</i>(β) is true for all <span class="nowrap">β < α</span>, then <i>P</i>(α) is true for <i>all</i> α. Or, more practically: in order to prove a property <i>P</i> for all ordinals α, one can assume that it is already known for all smaller <span class="nowrap">β < α</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Transfinite_recursion">Transfinite recursion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=9" title="Edit section: Transfinite recursion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by <a href="/wiki/Transfinite_recursion" class="mw-redirect" title="Transfinite recursion">transfinite recursion</a> – the proof that the result is well-defined uses transfinite induction. Let <i>F</i> denote a (class) function <i>F</i> to be defined on the ordinals. The idea now is that, in defining <i>F</i>(α) for an unspecified ordinal α, one may assume that <i>F</i>(β) is already defined for all <span class="nowrap">β < α</span> and thus give a formula for <i>F</i>(α) in terms of these <i>F</i>(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α. </p><p>Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function <i>F</i> by letting <i>F</i>(α) be the smallest ordinal not in the set <span class="nowrap">{<i>F</i>(β) | β < α}</span>, that is, the set consisting of all <i>F</i>(β) for <span class="nowrap">β < α</span>. This definition assumes the <i>F</i>(β) known in the very process of defining <i>F</i>; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact, <i>F</i>(0) makes sense since there is no ordinal <span class="nowrap">β < 0</span>, and the set <span class="nowrap">{<i>F</i>(β) | β < 0}</span> is empty. So <i>F</i>(0) is equal to 0 (the smallest ordinal of all). Now that <i>F</i>(0) is known, the definition applied to <i>F</i>(1) makes sense (it is the smallest ordinal not in the singleton set <span class="nowrap">{<i>F</i>(0)} = {0}</span>), and so on (the <i>and so on</i> is exactly transfinite induction). It turns out that this example is not very exciting, since provably <span class="nowrap"><i>F</i>(α) = α</span> for all ordinals α, which can be shown, precisely, by transfinite induction. </p> <div class="mw-heading mw-heading3"><h3 id="Successor_and_limit_ordinals">Successor and limit ordinals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=10" title="Edit section: Successor and limit ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a <i><a href="/wiki/Successor_ordinal" title="Successor ordinal">successor ordinal</a></i>, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α 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 \alpha \cup \{\alpha \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi>α</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \cup \{\alpha \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8c3fbbdc240872874a1be922e5150d930cfece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.883ex; height:2.843ex;" alt="{\displaystyle \alpha \cup \{\alpha \}}"></span> since its elements are those of α and α itself.<sup id="cite_ref-von_Neumann1923pp199-208_6-1" class="reference"><a href="#cite_note-von_Neumann1923pp199-208-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>A nonzero ordinal that is <i>not</i> a successor is called a <i><a href="/wiki/Limit_ordinal" title="Limit ordinal">limit ordinal</a></i>. One justification for this term is that a limit ordinal is the <a href="/wiki/Limit_point" class="mw-redirect" title="Limit point">limit</a> in a topological sense of all smaller ordinals (under the <a href="/wiki/Order_topology" title="Order topology">order topology</a>). </p><p>When <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 \langle \alpha _{\iota }|\iota <\gamma \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ι</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ι</mi> <mo><</mo> <mi>γ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \alpha _{\iota }|\iota <\gamma \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ac702e8241da1825c939999ba87f55495a3fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.942ex; height:2.843ex;" alt="{\displaystyle \langle \alpha _{\iota }|\iota <\gamma \rangle }"></span> is an ordinal-indexed sequence, indexed by a limit <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> and the sequence is <i>increasing</i>, i.e. <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 \alpha _{\iota }<\alpha _{\rho }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ι</mi> </mrow> </msub> <mo><</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{\iota }<\alpha _{\rho }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1859fb755d8542214c252fa89827ae449b23c45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.97ex; height:2.509ex;" alt="{\displaystyle \alpha _{\iota }<\alpha _{\rho }}"></span> whenever <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 \iota <\rho ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ι</mi> <mo><</mo> <mi>ρ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iota <\rho ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da8645bf8535f85dd9d96792b5ca4e4c1ed1162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.771ex; height:2.343ex;" alt="{\displaystyle \iota <\rho ,}"></span> its <i>limit</i> is defined as the least upper bound of the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\alpha _{\iota }|\iota <\gamma \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ι</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ι</mi> <mo><</mo> <mi>γ</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\alpha _{\iota }|\iota <\gamma \},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e68e9d0becdd8c5383e0b3f0d02590373d71841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.105ex; height:2.843ex;" alt="{\displaystyle \{\alpha _{\iota }|\iota <\gamma \},}"></span> that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals. </p><p>Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if: </p> <dl><dd>There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.</dd></dl> <p>So in the following sequence: </p> <dl><dd>0, 1, 2, ..., ω, ω+1</dd></dl> <p>ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω. </p><p>Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a function <i>F</i> by transfinite recursion on all ordinals, one defines <i>F</i>(0), and <i>F</i>(α+1) assuming <i>F</i>(α) is defined, and then, for limit ordinals δ one defines <i>F</i>(δ) as the limit of the <i>F</i>(β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit if <i>F</i> does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for <i>F</i> nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively). </p> <div class="mw-heading mw-heading3"><h3 id="Indexing_classes_of_ordinals">Indexing classes of ordinals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=11" title="Edit section: Indexing classes of ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any well-ordered set is similar (order-isomorphic) to a unique ordinal number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>; in other words, its elements can be indexed in increasing fashion by the ordinals less than <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span>. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span>. The same holds, with a slight modification, for <i>classes</i> of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>-th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. Formally, the definition is by transfinite induction: the <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>-th element of the class is defined (provided it has already been defined for all <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 \beta <\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo><</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta <\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d887fccd6816244086df6badf860ba816504b035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.693ex; height:2.676ex;" alt="{\displaystyle \beta <\gamma }"></span>), as the smallest element greater than the <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>-th element for all <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta <\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo><</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta <\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d887fccd6816244086df6badf860ba816504b035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.693ex; height:2.676ex;" alt="{\displaystyle \beta <\gamma }"></span>⁠</span>. </p><p>This could be applied, for example, to the class of limit ordinals: the <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>-th ordinal, which is either a limit or zero 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 \omega \cdot \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>⋅</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \cdot \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4435fb58e1feb747cd9ad1835b3b0b7b0748fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.176ex;" alt="{\displaystyle \omega \cdot \gamma }"></span> (see <a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">ordinal arithmetic</a> for the definition of multiplication of ordinals). Similarly, one can consider <i><a href="/wiki/Additively_indecomposable_ordinal" title="Additively indecomposable ordinal">additively indecomposable ordinals</a></i> (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>-th additively indecomposable ordinal is indexed as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a6f74d7256597b05dfd2a0d8924df2e177db74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.343ex;" alt="{\displaystyle \omega ^{\gamma }}"></span>⁠</span>. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>-th ordinal <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> such 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 \omega ^{\alpha }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\alpha }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858a06838aaa753a28eeb613da171e218f44704c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.316ex; height:2.343ex;" alt="{\displaystyle \omega ^{\alpha }=\alpha }"></span> is written <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2137e811accd25f099a39cf1bc58cbcdd3cb188a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.208ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{\gamma }}"></span>⁠</span>. These are called the "<a href="/wiki/Epsilon_numbers_(mathematics)" class="mw-redirect" title="Epsilon numbers (mathematics)">epsilon numbers</a>". </p> <div class="mw-heading mw-heading3"><h3 id="Closed_unbounded_sets_and_classes">Closed unbounded sets and classes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=12" title="Edit section: Closed unbounded sets and classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A class <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> of ordinals is said to be <i>unbounded</i>, or <i>cofinal</i>, when given any ordinal <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span>, there is 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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></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 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> such 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 \alpha <\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4250c976810fd3abc5916c47ee4984a7e5882b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \alpha <\beta }"></span> (then the class must be a proper class, i.e., it cannot be a set). It is said to be <i>closed</i> when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is continuous in the sense that, 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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> a limit ordinal, <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 F(\delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\delta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cef072d068e10c9668853feecb1f4728b36146d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.599ex; height:2.843ex;" alt="{\displaystyle F(\delta )}"></span> (the <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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>-th ordinal in the class) is the limit of all <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 F(\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f9db07b272b7034961ad583b0d8264989318b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.812ex; height:2.843ex;" alt="{\displaystyle F(\gamma )}"></span> 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 \gamma <\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo><</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma <\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e038c542d9770c4bf69b2fd8bedb00eee9bf89ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.409ex; height:2.843ex;" alt="{\displaystyle \gamma <\delta }"></span>; this is also the same as being closed, in the <a href="/wiki/Topological_space" title="Topological space">topological</a> sense, for the <a href="/wiki/Order_topology" title="Order topology">order topology</a> (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent). </p><p>Of particular importance are those classes of ordinals that are <a href="/wiki/Club_set" title="Club set">closed and unbounded</a>, sometimes called <i>clubs</i>. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class 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 \varepsilon _{\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/523f6066e848ffd999022488bcad941f23bb30d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.056ex; margin-bottom: -0.616ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{\cdot }}"></span> ordinals, or the class of <a href="#Ordinals_and_cardinals">cardinals</a>, are all closed unbounded; the set of <a href="#Cofinality">regular</a> cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded. </p><p>A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality. </p><p>Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>: A subset of a limit ordinal <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is said to be unbounded (or cofinal) under <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> provided any ordinal less than <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is less than some ordinal in the set. More generally, one can call a subset of any ordinal <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> cofinal 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> provided every ordinal less than <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is less than <i>or equal to</i> some ordinal in the set. The subset is said to be closed under <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> provided it is closed for the order topology <i>in</i> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span>, i.e. a limit of ordinals in the set is either in the set or equal to <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> itself. </p> <div class="mw-heading mw-heading2"><h2 id="Arithmetic_of_ordinals">Arithmetic of ordinals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=13" title="Edit section: Arithmetic of ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">Ordinal arithmetic</a></div> <p>There are three usual operations on ordinals: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion. The <a href="/wiki/Ordinal_arithmetic#Cantor_normal_form" title="Ordinal arithmetic">Cantor normal form</a> provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε<sub>0</sub> = ω<sup>ε<sub>0</sub></sup>. </p><p>Ordinals are a subclass of the class of <a href="/wiki/Surreal_number" title="Surreal number">surreal numbers</a>, and the so-called "natural" arithmetical operations for surreal numbers are an alternative way to combine ordinals arithmetically. They retain commutativity at the expense of continuity. </p><p>Interpreted as <i><a href="/wiki/Nimber" title="Nimber">nimbers</a></i>, a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations. These operations are commutative but the restriction to natural numbers is generally not the same as ordinary addition of natural numbers. </p> <div class="mw-heading mw-heading2"><h2 id="Ordinals_and_cardinals">Ordinals and cardinals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=14" title="Edit section: Ordinals and cardinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Initial_ordinal_of_a_cardinal">Initial ordinal of a cardinal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=15" title="Edit section: Initial ordinal of a cardinal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each ordinal associates with one <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal</a>, its cardinality. If there is a bijection between two ordinals (e.g. <span class="nowrap">ω = 1 + ω</span> and <span class="nowrap">ω + 1 > ω</span>), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called the <i>initial ordinal</i> of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. The <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the <a href="/wiki/Von_Neumann_cardinal_assignment" title="Von Neumann cardinal assignment">Von Neumann cardinal assignment</a> as the cardinal's representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (see <a href="/wiki/Scott%27s_trick" title="Scott's trick">Scott's trick</a>). </p><p>One issue with Scott's trick is that it identifies the cardinal number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> with <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\emptyset \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\emptyset \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2943085c6db53add570975fb7c528c398f0c52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{\emptyset \}}"></span>⁠</span>, which in some formulations is the ordinal number <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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 1}"></span>⁠</span>. It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers. </p><p>The α-th infinite initial ordinal is written <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f097b57783e051c7f975bb4cf2c20830ae0c6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.73ex; height:2.009ex;" alt="{\displaystyle \omega _{\alpha }}"></span>⁠</span>, it is always a limit ordinal. Its cardinality is written <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}"></span>⁠</span>. For example, the cardinality of ω<sub>0</sub> = ω is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>⁠</span>, which is also the cardinality of ω<sup>2</sup> or ε<sub>0</sub> (all are countable ordinals). So ω can be identified with <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>⁠</span>, except that the notation <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 \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span> is used when writing cardinals, and ω when writing ordinals (this is important since, for example, <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 \aleph _{0}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6810b7dffe2e82c6417d9826a86d23abe16a027f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.475ex; height:3.176ex;" alt="{\displaystyle \aleph _{0}^{2}}"></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 \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span> whereas <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 \omega ^{2}>\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}>\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f276be4f7f2f77195c377e4b3897085a8ece69be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.044ex; height:2.676ex;" alt="{\displaystyle \omega ^{2}>\omega }"></span>). Also, <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 \omega _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}"></span> is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, 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 \omega _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}"></span> is the order type of that 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 \omega _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b914a8bfef5d1b9b106048afa0aab4a99251f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}"></span> is the smallest ordinal whose cardinality is greater than <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>⁠</span>, and so on, 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 \omega _{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649576b619f8aad6c47a90ca6266c03c4accc481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.7ex; height:2.009ex;" alt="{\displaystyle \omega _{\omega }}"></span> is the limit of the <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 \omega _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57263f565851485af54c589561735513ac456858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.664ex; height:2.009ex;" alt="{\displaystyle \omega _{n}}"></span> for natural numbers <i>n</i> (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the <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 \omega _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57263f565851485af54c589561735513ac456858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.664ex; height:2.009ex;" alt="{\displaystyle \omega _{n}}"></span>). </p> <div class="mw-heading mw-heading3"><h3 id="Cofinality">Cofinality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=16" title="Edit section: Cofinality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Cofinality" title="Cofinality">cofinality</a> of an ordinal <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is the smallest ordinal <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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> that is the order type of a <a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">cofinal</a> subset of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span>. Notice that a number of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set. </p><p>Thus for a limit ordinal, there exists 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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>-indexed strictly increasing sequence with limit <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span>. For example, the cofinality of ω<sup>2</sup> is ω, because the sequence ω·<i>m</i> (where <i>m</i> ranges over the natural numbers) tends to ω<sup>2</sup>; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does <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 \omega _{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649576b619f8aad6c47a90ca6266c03c4accc481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.7ex; height:2.009ex;" alt="{\displaystyle \omega _{\omega }}"></span> or an uncountable cofinality. </p><p>The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>⁠</span>. </p><p>An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom of Choice, 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 \omega _{\alpha +1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\alpha +1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff08a9e70810bc6e66e7a719451ef1d0f2f297a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.831ex; height:2.009ex;" alt="{\displaystyle \omega _{\alpha +1}}"></span> is regular for each α. In this case, the ordinals 0, 1, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>⁠</span>, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}"></span>⁠</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 \omega _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b914a8bfef5d1b9b106048afa0aab4a99251f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}"></span> are regular, whereas 2, 3, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649576b619f8aad6c47a90ca6266c03c4accc481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.7ex; height:2.009ex;" alt="{\displaystyle \omega _{\omega }}"></span>⁠</span>, and ω<sub>ω·2</sub> are initial ordinals that are not regular. </p><p>The cofinality of any ordinal <i>α</i> is a regular ordinal, i.e. the cofinality of the cofinality of <i>α</i> is the same as the cofinality of <i>α</i>. So the cofinality operation is <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Some_"large"_countable_ordinals"><span id="Some_.22large.22_countable_ordinals"></span>Some "large" countable ordinals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=17" title="Edit section: Some "large" countable ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Large_countable_ordinal" title="Large countable ordinal">Large countable ordinal</a></div> <p>As mentioned above (see <a href="/wiki/Ordinal_arithmetic#Cantor_normal_form" title="Ordinal arithmetic">Cantor normal form</a>), the ordinal ε<sub>0</sub> is the smallest satisfying the equation <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\alpha }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\alpha }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858a06838aaa753a28eeb613da171e218f44704c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.316ex; height:2.343ex;" alt="{\displaystyle \omega ^{\alpha }=\alpha }"></span>⁠</span>, so it is the limit of the sequence 0, 1, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>⁠</span>, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa5636cabcea62c44c8b91fd9095e06054a6fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.7ex; height:2.343ex;" alt="{\displaystyle \omega ^{\omega }}"></span>⁠</span>, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\omega ^{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\omega ^{\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e322088d40a99bd8b5540c3c0fdaffced1cc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.695ex; height:2.676ex;" alt="{\displaystyle \omega ^{\omega ^{\omega }}}"></span>⁠</span>, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the <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 \iota }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ι</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iota }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce48dd56254d0a7c33e987c7c8eeb44c963ac04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.823ex; height:1.676ex;" alt="{\displaystyle \iota }"></span>-th ordinal such 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 \omega ^{\alpha }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\alpha }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858a06838aaa753a28eeb613da171e218f44704c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.316ex; height:2.343ex;" alt="{\displaystyle \omega ^{\alpha }=\alpha }"></span> is called <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\iota }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ι</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\iota }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d6b1688fbdbcb542466ef547dc77f6bae8bda45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.898ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{\iota }}"></span>⁠</span>, then one could go on trying to find the <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 \iota }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ι</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iota }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce48dd56254d0a7c33e987c7c8eeb44c963ac04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.823ex; height:1.676ex;" alt="{\displaystyle \iota }"></span>-th ordinal such that <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\alpha }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\alpha }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7698ee54562612764c71a2b0da16e68c3122c032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.954ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{\alpha }=\alpha }"></span>⁠</span>, "and so on", but all the subtlety lies in the "and so on"). One could try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limits a system of construction in this manner is the <a href="/wiki/Church%E2%80%93Kleene_ordinal" class="mw-redirect" title="Church–Kleene ordinal">Church–Kleene ordinal</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 \omega _{1}^{\mathrm {CK} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">K</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}^{\mathrm {CK} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0287c1265a806f47a4187ef41cec17113c53f204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.143ex; height:3.176ex;" alt="{\displaystyle \omega _{1}^{\mathrm {CK} }}"></span> (despite the <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 \omega _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}"></span> in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by a <a href="/wiki/Computable_function" title="Computable function">computable function</a> (this can be made rigorous, of course). Considerably large ordinals can be defined below <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}^{\mathrm {CK} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">K</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}^{\mathrm {CK} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0287c1265a806f47a4187ef41cec17113c53f204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.143ex; height:3.176ex;" alt="{\displaystyle \omega _{1}^{\mathrm {CK} }}"></span>⁠</span>, however, which measure the "proof-theoretic strength" of certain <a href="/wiki/Formal_system" title="Formal system">formal systems</a> (for example, <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 \varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb0a8377db20e42274444cb181d51b5532b5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}"></span> measures the strength of <a href="/wiki/Peano_axioms" title="Peano axioms">Peano arithmetic</a>). Large countable ordinals such as countable <a href="/wiki/Admissible_ordinal" title="Admissible ordinal">admissible ordinals</a> can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.<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. (November 2019)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Topology_and_ordinals">Topology and ordinals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=18" title="Edit section: Topology and ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Order_topology" title="Order topology">Order topology</a></div> <p>Any ordinal number can be made into a <a href="/wiki/Topological_space" title="Topological space">topological space</a> by endowing it with the <a href="/wiki/Order_topology" title="Order topology">order topology</a>; this topology is <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete</a> if and only if it is less than or equal to ω. A subset of ω + 1 is open in the order topology if and only if either it is <a href="/wiki/Cofinite" class="mw-redirect" title="Cofinite">cofinite</a> or it does not contain ω as an element. </p><p>See the <a href="/wiki/Order_topology#Topology_and_ordinals" title="Order topology">Topology and ordinals</a> section of the "Order topology" article. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=19" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The transfinite ordinal numbers, which first appeared in 1883,<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> originated in Cantor's work with <a href="/wiki/Derived_set_(mathematics)" title="Derived set (mathematics)">derived sets</a>. If <i>P</i> is a set of real numbers, the derived set <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is the set of <a href="/wiki/Limit_point" class="mw-redirect" title="Limit point">limit points</a> of <i>P</i>. In 1872, Cantor generated the sets <i>P</i><sup>(<i>n</i>)</sup> by applying the derived set operation <i>n</i> times to <i>P</i>. In 1880, he pointed out that these sets form the sequence <span class="texhtml"><i>P' </i>⊇ ··· ⊇ <i>P</i><sup>(<i>n</i>)</sup> ⊇ <i>P</i><sup>(<i>n</i> + 1)</sup> ⊇ ···,</span> and he continued the derivation process by defining <i>P</i><sup>(∞)</sup> as the intersection of these sets. Then he iterated the derived set operation and intersections to extend his sequence of sets into the infinite: <span class="texhtml"><i>P</i><sup>(∞)</sup> ⊇ <i>P</i><sup>(∞ + 1)</sup> ⊇ <i>P</i><sup>(∞ + 2)</sup> ⊇ ··· ⊇ <i>P</i><sup>(2∞)</sup> ⊇ ··· ⊇ <i>P</i><sup>(∞<sup>2</sup>)</sup> ⊇ ···.</span><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> The superscripts containing ∞ are just indices defined by the derivation process.<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>Cantor used these sets in the theorems: </p> <div><ol><li>If <span class="texhtml"><i>P</i><sup>(<i>α</i>)</sup> = ∅</span> for some index <i>α</i>, then <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is countable;</li><li>Conversely, if <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is countable, then there is an index <i>α</i> such that <span class="texhtml"><i>P</i><sup>(<i>α</i>)</sup> = ∅</span>.</li></ol></div> <p>These theorems are proved by partitioning <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> into <a href="/wiki/Pairwise_disjoint" class="mw-redirect" title="Pairwise disjoint">pairwise disjoint</a> sets: <span class="texhtml"><i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> = (<i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i>\ <i>P</i><sup>(2)</sup>) ∪ (<i>P</i><sup>(2)</sup> \ <i>P</i><sup>(3)</sup>) ∪ ··· ∪ (<i>P</i><sup>(∞)</sup> \ <i>P</i><sup>(∞ + 1)</sup>) ∪ ··· ∪ <i>P</i><sup>(<i>α</i>)</sup></span>. For <span class="texhtml"><i>β</i> < <i>α</i>:</span> since <span class="texhtml"><i>P</i><sup>(<i>β</i> + 1)</sup></span> contains the limit points of <i>P</i><sup>(<i>β</i>)</sup>, the sets <span class="texhtml"><i>P</i><sup>(<i>β</i>)</sup> \ <i>P</i><sup>(<i>β</i> + 1)</sup></span> have no limit points. Hence, they are <a href="/wiki/Discrete_set" class="mw-redirect" title="Discrete set">discrete sets</a>, so they are countable. Proof of first theorem: If <span class="texhtml"><i>P</i><sup>(<i>α</i>)</sup> = ∅</span> for some index <i>α</i>, then <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is the countable union of countable sets. Therefore, <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is countable.<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 second theorem requires proving the existence of an <i>α</i> such that <span class="texhtml"><i>P</i><sup>(<i>α</i>)</sup> = ∅</span>. To prove this, Cantor considered the set of all <i>α</i> having countably many predecessors. To define this set, he defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ with <i>ω</i>, the first transfinite ordinal number. Cantor called the set of finite ordinals the first <b>number class</b>. The second number class is the set of ordinals whose predecessors form a countably infinite set. The set of all <i>α</i> having countably many predecessors—that is, the set of countable ordinals—is the union of these two number classes. Cantor proved that the cardinality of the second number class is the first uncountable cardinality.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Cantor's second theorem becomes: If <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is countable, then there is a countable ordinal <i>α</i> such that <span class="texhtml"><i>P</i><sup>(<i>α</i>)</sup> = ∅</span>. Its proof uses <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a>. Let <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> be countable, and assume there is no such α. This assumption produces two cases. </p> <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="margin-left: 1.6em;"> <ul><li>Case 1: <span class="texhtml"><i>P</i><sup>(<i>β</i>)</sup> \ <i>P</i><sup>(<i>β</i> + 1)</sup></span> is non-empty for all countable <i>β</i>. Since there are uncountably many of these pairwise disjoint sets, their union is uncountable. This union is a subset of <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i>, so <i>P' </i> is uncountable.</li> <li>Case 2: <span class="texhtml"><i>P</i><sup>(<i>β</i>)</sup> \ <i>P</i><sup>(<i>β</i> + 1)</sup></span> is empty for some countable <i>β</i>. Since <span class="texhtml"><i>P</i><sup>(<i>β</i> + 1)</sup> ⊆ <i>P</i><sup>(<i>β</i>)</sup></span>, this implies <span class="texhtml"><i>P</i><sup>(<i>β</i> + 1)</sup> = <i>P</i><sup>(<i>β</i>)</sup></span>. Thus, <i>P</i><sup>(<i>β</i>)</sup> is a <a href="/wiki/Derived_set_(mathematics)#Properties" title="Derived set (mathematics)">perfect set</a>, so it is uncountable.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Since <span class="texhtml"><i>P</i><sup>(<i>β</i>)</sup> ⊆ <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i></span>, the set <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is uncountable.</li></ul> </div> <p>In both cases, <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> is uncountable, which contradicts <i>P<span class="nowrap" style="padding-left:0.05em;">′</span></i> being countable. Therefore, there is a countable ordinal <i>α</i> such that <span class="texhtml"><i>P</i><sup>(<i>α</i>)</sup> = ∅</span>. Cantor's work with derived sets and ordinal numbers led to the <a href="/wiki/Cantor-Bendixson_theorem" class="mw-redirect" title="Cantor-Bendixson theorem">Cantor-Bendixson theorem</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Using successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The <span class="texhtml">(<i>α</i> + 1)</span>-th number class is the set of ordinals whose predecessors form a set of the same cardinality as the <i>α</i>-th number class. The cardinality of the <span class="texhtml">(<i>α</i> + 1)</span>-th number class is the cardinality immediately following that of the <i>α</i>-th number class.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> For a limit ordinal <i>α</i>, the <i>α</i>-th number class is the union of the <i>β</i>-th number classes for <span class="texhtml"><i>β</i> < <i>α</i></span>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Its cardinality is the limit of the cardinalities of these number classes. </p><p>If <i>n</i> is finite, the <i>n</i>-th number class has cardinality <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4f324a84c66831b420afd97b76e07850fe2ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.739ex; height:2.509ex;" alt="{\displaystyle \aleph _{n-1}}"></span>⁠</span>. If <span class="texhtml"><i>α</i> ≥ <i>ω</i></span>, the <i>α</i>-th number class has cardinality <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}"></span>⁠</span>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Therefore, the cardinalities of the number classes correspond one-to-one with the <a href="/wiki/Aleph_number" title="Aleph number">aleph numbers</a>. Also, the <i>α</i>-th number class consists of ordinals different from those in the preceding number classes if and only if <i>α</i> is a non-limit ordinal. Therefore, the non-limit number classes partition the ordinals into pairwise disjoint sets. </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=Ordinal_number&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Counting" title="Counting">Counting</a></li> <li><a href="/wiki/Even_and_odd_ordinals" title="Even and odd ordinals">Even and odd ordinals</a></li> <li><a href="/wiki/First_uncountable_ordinal" title="First uncountable ordinal">First uncountable ordinal</a></li> <li><a href="/wiki/Order_topology#Ordinal_space" title="Order topology">Ordinal space</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal number</a>, a generalization of ordinals which includes negative, real, and infinitesimal values.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ordinal_number&action=edit&section=21" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://literarydevices.net/ordinal-number/">"Ordinal Number - Examples and Definition of Ordinal Number"</a>. <i>Literary Devices</i>. 2017-05-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-08-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Literary+Devices&rft.atitle=Ordinal+Number+-+Examples+and+Definition+of+Ordinal+Number&rft.date=2017-05-21&rft_id=https%3A%2F%2Fliterarydevices.net%2Fordinal-number%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSterling2007" class="citation book cs1">Sterling, Kristin (2007-09-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h1sGybxETuQC"><i>Ordinal Numbers</i></a>. LernerClassroom. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8225-8846-7" title="Special:BookSources/978-0-8225-8846-7"><bdi>978-0-8225-8846-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinal+Numbers&rft.pub=LernerClassroom&rft.date=2007-09-01&rft.isbn=978-0-8225-8846-7&rft.aulast=Sterling&rft.aufirst=Kristin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dh1sGybxETuQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" 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">Thorough introductions are given by <a href="#CITEREFLevy1979">Levy 1979</a> and <a href="#CITEREFJech2003">Jech 2003</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallett1979" class="citation cs2">Hallett, Michael (1979), "Towards a theory of mathematical research programmes. I", <i>The British Journal for the Philosophy of Science</i>, <b>30</b> (1): 1–25, <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%2Fbjps%2F30.1.1">10.1093/bjps/30.1.1</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0532548">0532548</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+British+Journal+for+the+Philosophy+of+Science&rft.atitle=Towards+a+theory+of+mathematical+research+programmes.+I&rft.volume=30&rft.issue=1&rft.pages=1-25&rft.date=1979&rft_id=info%3Adoi%2F10.1093%2Fbjps%2F30.1.1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D532548%23id-name%3DMR&rft.aulast=Hallett&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>. See the footnote on p. 12.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/OrdinalNumber.html">"Ordinal Number"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Ordinal+Number&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FOrdinalNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span></span> </li> <li id="cite_note-von_Neumann1923pp199-208-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-von_Neumann1923pp199-208_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-von_Neumann1923pp199-208_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFvon_Neumann1923">von Neumann 1923</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFLevy1979">Levy 1979</a>, p. 52 attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s.</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">Cantor 1883. English translation: <a href="#CITEREFEwald1996">Ewald 1996</a>, pp. 881–920</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"><a href="#CITEREFFerreirós1995">Ferreirós 1995</a>, pp. 34–35; <a href="#CITEREFFerreirós2007">Ferreirós 2007</a>, pp. 159, 204–5</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"><a href="#CITEREFFerreirós2007">Ferreirós 2007</a>, p. 269</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"><a href="#CITEREFFerreirós1995">Ferreirós 1995</a>, pp. 35–36; <a href="#CITEREFFerreirós2007">Ferreirós 2007</a>, p. 207</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFFerreirós1995">Ferreirós 1995</a>, pp. 36–37; <a href="#CITEREFFerreirós2007">Ferreirós 2007</a>, p. 271</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"><a href="#CITEREFDauben1979">Dauben 1979</a>, p. 111</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"><a href="#CITEREFFerreirós2007">Ferreirós 2007</a>, pp. 207–8</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"><a href="#CITEREFDauben1979">Dauben 1979</a>, pp. 97–98</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"><a href="#CITEREFHallett1986">Hallett 1986</a>, pp. 61–62</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"><a href="#CITEREFTait1997">Tait 1997</a>, p. 5 footnote</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">The first number class has cardinality <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>⁠</span>. <a href="/wiki/Mathematical_induction" title="Mathematical induction">Mathematical induction</a> proves that the <i>n</i>-th number class has cardinality <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4f324a84c66831b420afd97b76e07850fe2ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.739ex; height:2.509ex;" alt="{\displaystyle \aleph _{n-1}}"></span>⁠</span>. Since the <i>ω</i>-th number class is the union of the <i>n</i>-th number classes, its cardinality is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab056732a610ab69363c5e2bd54dfe0217d1883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.675ex; height:2.509ex;" alt="{\displaystyle \aleph _{\omega }}"></span>⁠</span>, the limit of the <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4f324a84c66831b420afd97b76e07850fe2ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.739ex; height:2.509ex;" alt="{\displaystyle \aleph _{n-1}}"></span>⁠</span>. <a href="#Transfinite_induction">Transfinite induction</a> proves that if <span class="texhtml"><i>α</i> ≥ <i>ω</i></span>, the <i>α</i>-th number class has cardinality <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}"></span>⁠</span>.</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=Ordinal_number&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCantor1883" class="citation cs2"><a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor, Georg</a> (1883), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0021&DMDID=DMDLOG_0051">"Ueber unendliche, lineare Punktmannichfaltigkeiten. 5."</a>, <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>21</b> (4): 545–591, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01446819">10.1007/bf01446819</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121930608">121930608</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Ueber+unendliche%2C+lineare+Punktmannichfaltigkeiten.+5.&rft.volume=21&rft.issue=4&rft.pages=545-591&rft.date=1883&rft_id=info%3Adoi%2F10.1007%2Fbf01446819&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121930608%23id-name%3DS2CID&rft.aulast=Cantor&rft.aufirst=Georg&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN235181684_0021%26DMDID%3DDMDLOG_0051&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>. Published separately as: <i>Grundlagen einer allgemeinen Mannigfaltigkeitslehre</i>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCantor1897" class="citation cs2">Cantor, Georg (1897), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428224">"Beitrage zur Begrundung der transfiniten Mengenlehre. II"</a>, <i>Mathematische Annalen</i>, vol. 49, no. 2, pp. 207–246, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01444205">10.1007/BF01444205</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121665994">121665994</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Beitrage+zur+Begrundung+der+transfiniten+Mengenlehre.+II&rft.volume=49&rft.issue=2&rft.pages=207-246&rft.date=1897&rft_id=info%3Adoi%2F10.1007%2FBF01444205&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121665994%23id-name%3DS2CID&rft.aulast=Cantor&rft.aufirst=Georg&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1428224&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.org/details/117770262">English translation: Contributions to the Founding of the Theory of Transfinite Numbers II</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwayGuy2012" class="citation cs2"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John H.</a>; <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard</a> (2012) [1996], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rfLSBwAAQBAJ&pg=PA266">"Cantor's Ordinal Numbers"</a>, <i>The Book of Numbers</i>, Springer, pp. 266–7, 274, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-4072-3" title="Special:BookSources/978-1-4612-4072-3"><bdi>978-1-4612-4072-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Cantor%27s+Ordinal+Numbers&rft.btitle=The+Book+of+Numbers&rft.pages=266-7%2C+274&rft.pub=Springer&rft.date=2012&rft.isbn=978-1-4612-4072-3&rft.aulast=Conway&rft.aufirst=John+H.&rft.au=Guy%2C+Richard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrfLSBwAAQBAJ%26pg%3DPA266&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDauben1979" class="citation cs2"><a href="/wiki/Joseph_Dauben" title="Joseph Dauben">Dauben, Joseph</a> (1979), <i>Georg Cantor: His Mathematics and Philosophy of the Infinite</i>, <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-34871-0" title="Special:BookSources/0-674-34871-0"><bdi>0-674-34871-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Georg+Cantor%3A+His+Mathematics+and+Philosophy+of+the+Infinite&rft.pub=Harvard+University+Press&rft.date=1979&rft.isbn=0-674-34871-0&rft.aulast=Dauben&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEwald1996" class="citation cs2">Ewald, William B., ed. (1996), <i>From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2</i>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850536-1" title="Special:BookSources/0-19-850536-1"><bdi>0-19-850536-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Immanuel+Kant+to+David+Hilbert%3A+A+Source+Book+in+the+Foundations+of+Mathematics%2C+Volume+2&rft.pub=Oxford+University+Press&rft.date=1996&rft.isbn=0-19-850536-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerreirós1995" class="citation cs2">Ferreirós, José (1995), <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/17322/1/HMat1995.pdf">"<span class="cs1-kern-left"></span>'What fermented in me for years': Cantor's discovery of transfinite numbers"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a></i>, <b>22</b>: 33–42, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.1995.1003">10.1006/hmat.1995.1003</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=%27What+fermented+in+me+for+years%27%3A+Cantor%27s+discovery+of+transfinite+numbers&rft.volume=22&rft.pages=33-42&rft.date=1995&rft_id=info%3Adoi%2F10.1006%2Fhmat.1995.1003&rft.aulast=Ferreir%C3%B3s&rft.aufirst=Jos%C3%A9&rft_id=http%3A%2F%2Fphilsci-archive.pitt.edu%2F17322%2F1%2FHMat1995.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerreirós2007" class="citation cs2">Ferreirós, José (2007), <i>Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought</i> (2nd revised ed.), <a href="/wiki/Birkh%C3%A4user" title="Birkhäuser">Birkhäuser</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-8349-7" title="Special:BookSources/978-3-7643-8349-7"><bdi>978-3-7643-8349-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Labyrinth+of+Thought%3A+A+History+of+Set+Theory+and+Its+Role+in+Mathematical+Thought&rft.edition=2nd+revised&rft.pub=Birkh%C3%A4user&rft.date=2007&rft.isbn=978-3-7643-8349-7&rft.aulast=Ferreir%C3%B3s&rft.aufirst=Jos%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallett1986" class="citation cs2">Hallett, Michael (1986), <i>Cantorian Set Theory and Limitation of Size</i>, Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853283-0" title="Special:BookSources/0-19-853283-0"><bdi>0-19-853283-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cantorian+Set+Theory+and+Limitation+of+Size&rft.pub=Oxford+University+Press&rft.date=1986&rft.isbn=0-19-853283-0&rft.aulast=Hallett&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1982" class="citation cs2">Hamilton, A. G. (1982), "6. Ordinal and cardinal numbers", <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/numberssetsaxiom0000hami"><i>Numbers, Sets, and Axioms : the Apparatus of Mathematics</i></a></span>, New York: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-24509-5" title="Special:BookSources/0-521-24509-5"><bdi>0-521-24509-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=6.+Ordinal+and+cardinal+numbers&rft.btitle=Numbers%2C+Sets%2C+and+Axioms+%3A+the+Apparatus+of+Mathematics&rft.place=New+York&rft.pub=Cambridge+University+Press&rft.date=1982&rft.isbn=0-521-24509-5&rft.aulast=Hamilton&rft.aufirst=A.+G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnumberssetsaxiom0000hami&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKanamori2012" class="citation cs2"><a href="/wiki/Akihiro_Kanamori" title="Akihiro Kanamori">Kanamori, Akihiro</a> (2012), <a rel="nofollow" class="external text" href="http://math.bu.edu/people/aki/16.pdf">"Set Theory from Cantor to Cohen"</a> <span class="cs1-format">(PDF)</span>, in Gabbay, Dov M.; Kanamori, Akihiro; Woods, John H. (eds.), <i>Sets and Extensions in the Twentieth Century</i>, Cambridge University Press, pp. 1–71, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51621-3" title="Special:BookSources/978-0-444-51621-3"><bdi>978-0-444-51621-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Set+Theory+from+Cantor+to+Cohen&rft.btitle=Sets+and+Extensions+in+the+Twentieth+Century&rft.pages=1-71&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=978-0-444-51621-3&rft.aulast=Kanamori&rft.aufirst=Akihiro&rft_id=http%3A%2F%2Fmath.bu.edu%2Fpeople%2Faki%2F16.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevy1979" class="citation cs2">Levy, A. (2002) [1979], <i>Basic Set Theory</i>, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-42079-5" title="Special:BookSources/0-486-42079-5"><bdi>0-486-42079-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Set+Theory&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=0-486-42079-5&rft.aulast=Levy&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJech2003" class="citation cs2">Jech, Thomas (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GHjmCAAAQBAJ"><i>Set Theory</i></a> (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-22400-7" title="Special:BookSources/978-3-662-22400-7"><bdi>978-3-662-22400-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory&rft.edition=2nd&rft.pub=Springer&rft.date=2013&rft.isbn=978-3-662-22400-7&rft.aulast=Jech&rft.aufirst=Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGHjmCAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSierpiński1965" class="citation cs2">Sierpiński, W. (1965), <a href="/wiki/Cardinal_and_Ordinal_Numbers" title="Cardinal and Ordinal Numbers"><i>Cardinal and Ordinal Numbers</i></a> (2nd ed.), Warszawa: Państwowe Wydawnictwo Naukowe</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cardinal+and+Ordinal+Numbers&rft.place=Warszawa&rft.edition=2nd&rft.pub=Pa%C5%84stwowe+Wydawnictwo+Naukowe&rft.date=1965&rft.aulast=Sierpi%C5%84ski&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span> Also defines ordinal operations in terms of the Cantor Normal Form.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuppes1960" class="citation cs2"><a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes, Patrick</a> (1960), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/axiomaticsettheo00supp_0"><i>Axiomatic Set Theory</i></a></span>, D.Van Nostrand, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61630-4" title="Special:BookSources/0-486-61630-4"><bdi>0-486-61630-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Axiomatic+Set+Theory&rft.pub=D.Van+Nostrand&rft.date=1960&rft.isbn=0-486-61630-4&rft.aulast=Suppes&rft.aufirst=Patrick&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faxiomaticsettheo00supp_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTait1997" class="citation cs2"><a href="/wiki/William_W._Tait" title="William W. Tait">Tait, William W.</a> (1997), <a rel="nofollow" class="external text" href="http://home.uchicago.edu/wwtx/frege.cantor.dedekind.pdf">"Frege versus Cantor and Dedekind: On the Concept of Number"</a> <span class="cs1-format">(PDF)</span>, in William W. Tait (ed.), <i>Early Analytic Philosophy: Frege, Russell, Wittgenstein</i>, Open Court, pp. 213–248, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8126-9344-2" title="Special:BookSources/0-8126-9344-2"><bdi>0-8126-9344-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Frege+versus+Cantor+and+Dedekind%3A+On+the+Concept+of+Number&rft.btitle=Early+Analytic+Philosophy%3A+Frege%2C+Russell%2C+Wittgenstein&rft.pages=213-248&rft.pub=Open+Court&rft.date=1997&rft.isbn=0-8126-9344-2&rft.aulast=Tait&rft.aufirst=William+W.&rft_id=http%3A%2F%2Fhome.uchicago.edu%2Fwwtx%2Ffrege.cantor.dedekind.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1923" class="citation cs2"><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (1923), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141218090535/http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=">"Zur Einführung der transfiniten Zahlen"</a>, <i>Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum</i>, vol. 1, pp. 199–208, archived from <a rel="nofollow" class="external text" href="http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=">the original</a> on 2014-12-18<span class="reference-accessdate">, retrieved <span class="nowrap">2013-09-15</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+litterarum+ac+scientiarum+Ragiae+Universitatis+Hungaricae+Francisco-Josephinae%2C+Sectio+scientiarum+mathematicarum&rft.atitle=Zur+Einf%C3%BChrung+der+transfiniten+Zahlen&rft.volume=1&rft.pages=199-208&rft.date=1923&rft.aulast=von+Neumann&rft.aufirst=John&rft_id=http%3A%2F%2Facta.fyx.hu%2Facta%2FshowCustomerArticle.action%3Fid%3D4981%26dataObjectType%3Darticle%26returnAction%3DshowCustomerVolume%26sessionDataSetId%3D39716d660ae98d02%26style%3D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann2002" class="citation cs2"><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (January 2002) [1923], <a rel="nofollow" class="external text" href="http://www.hup.harvard.edu/catalog.php?isbn=9780674324497">"On the introduction of transfinite numbers"</a>, in Jean van Heijenoort (ed.), <i>From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931</i> (3rd ed.), Harvard University Press, pp. 346–354, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-32449-8" title="Special:BookSources/0-674-32449-8"><bdi>0-674-32449-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+introduction+of+transfinite+numbers&rft.btitle=From+Frege+to+G%C3%B6del%3A+A+Source+Book+in+Mathematical+Logic%2C+1879%E2%80%931931&rft.pages=346-354&rft.edition=3rd&rft.pub=Harvard+University+Press&rft.date=2002-01&rft.isbn=0-674-32449-8&rft.aulast=von+Neumann&rft.aufirst=John&rft_id=http%3A%2F%2Fwww.hup.harvard.edu%2Fcatalog.php%3Fisbn%3D9780674324497&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span> - English translation of <a href="#CITEREFvon_Neumann1923">von Neumann 1923</a>.</li></ul> </div> <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=Ordinal_number&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 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><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/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/ordinal" class="extiw" title="wiktionary:ordinal">ordinal</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/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Ordinal_numbers" class="extiw" title="commons:Category:Ordinal numbers">Ordinal numbers</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Ordinal_number">"Ordinal number"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ordinal+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DOrdinal_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrdinal+number" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.apronus.com/provenmath/ordinals.htm">Ordinals at ProvenMath</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220121063447/http://www.mtnmath.com/ord/index.html">Ordinal calculator</a> <a href="/wiki/GNU_General_Public_License" title="GNU General Public License">GPL'd</a> free software for computing with ordinals and ordinal notations</li> <li>Chapter 4 of <a rel="nofollow" class="external text" href="http://math.colorado.edu/~monkd">Don Monk's</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150429210249/http://euclid.colorado.edu/~monkd/m8714.html">lecture notes</a> on set theory is an introduction to ordinals.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline 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href="/wiki/Template:Countable_ordinals" title="Template:Countable ordinals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Countable_ordinals" title="Template talk:Countable ordinals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Countable_ordinals" title="Special:EditPage/Template:Countable ordinals"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Large_countable_ordinals" style="font-size:114%;margin:0 4em"><a href="/wiki/Large_countable_ordinal" title="Large countable ordinal">Large countable ordinals</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Omega_(ordinal)" class="mw-redirect" title="Omega (ordinal)">First infinite ordinal</a> <i>ω</i></li> <li><a href="/wiki/Epsilon_number" title="Epsilon number">Epsilon numbers</a> <i>ε</i><sub>0</sub></li> <li><a href="/wiki/Feferman%E2%80%93Sch%C3%BCtte_ordinal" title="Feferman–Schütte ordinal">Feferman–Schütte ordinal</a> Γ<sub>0</sub></li> <li><a href="/wiki/Ackermann_ordinal" title="Ackermann ordinal">Ackermann ordinal</a> <i>θ</i>(Ω<sup>2</sup>)</li> <li><a href="/wiki/Small_Veblen_ordinal" title="Small Veblen ordinal">small Veblen ordinal</a> <i>θ</i>(Ω<sup><i>ω</i></sup>)</li> <li><a href="/wiki/Large_Veblen_ordinal" title="Large Veblen ordinal">large Veblen ordinal</a> <i>θ</i>(Ω<sup>Ω</sup>)</li> <li><a href="/wiki/Bachmann%E2%80%93Howard_ordinal" title="Bachmann–Howard ordinal">Bachmann–Howard ordinal</a> <i>ψ</i>(<i>ε</i><sub>Ω+1</sub>)</li> <li><a href="/wiki/Buchholz%27s_ordinal" title="Buchholz's ordinal">Buchholz's ordinal</a> <i>ψ</i><sub>0</sub>(Ω<sub><i>ω</i></sub>)</li> <li><a href="/wiki/Takeuti%E2%80%93Feferman%E2%80%93Buchholz_ordinal" title="Takeuti–Feferman–Buchholz ordinal">Takeuti–Feferman–Buchholz ordinal</a> <i>ψ</i>(<i>ε</i><sub>Ω<sub><i>ω</i></sub>+1</sub>)</li> <li>Proof-theoretic ordinals of <a href="/wiki/Theories_of_iterated_inductive_definitions" title="Theories of iterated inductive definitions">the theories of iterated inductive definitions</a></li> <li><a href="/wiki/Computable_ordinal" title="Computable ordinal">Computable ordinals</a> < ω‍<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">CK</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span></li> <li><a href="/wiki/Nonrecursive_ordinal" title="Nonrecursive ordinal">Nonrecursive ordinal</a> ≥ ω‍<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">CK</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span></li> <li><a href="/wiki/First_uncountable_ordinal" title="First uncountable ordinal">First uncountable ordinal</a> <i>Ω</i></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Number_systems" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_systems" title="Template:Number systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_systems" title="Template talk:Number systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_systems" title="Special:EditPage/Template:Number systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</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/Natural_number" title="Natural number">Natural numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {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>)</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" 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 {A} }"></span>)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</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 {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</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/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br />types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a><br /> Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</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/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }"></span>)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</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/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a class="mw-selflink selflink">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</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/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Number#Main_classification" title="Number">Classification</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"></div><div role="navigation" class="navbox" aria-labelledby="Set_theory" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Set_theory" title="Template:Set theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Set_theory" title="Template talk:Set theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Set_theory" title="Special:EditPage/Template:Set theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Set_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</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/Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Venn_diagram" title="Venn diagram"><img alt="Venn diagram of set intersection" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/100px-Venn_A_intersect_B.svg.png" decoding="async" width="100" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/150px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/200px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Axiom" title="Axiom">Axioms</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/Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li> <li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">Choice</a> <ul><li><a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li> <li><a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li> <li><a href="/wiki/Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a> <ul><li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">projective</a></li></ul></li> <li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li> <li><a href="/wiki/Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Union</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li></ul> <ul><li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> <ul><li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li> <li><a href="/wiki/Power_set" title="Power set">Power set</a></li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost" title="Almost">Almost</a></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="/wiki/Large_cardinal" title="Large cardinal">large</a>)</li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li><a href="/wiki/Constructible_universe" title="Constructible universe">Constructible universe</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li> <li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a> <ul><li><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></li> <li><a href="/wiki/Tuple" title="Tuple">tuple</a></li></ul></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Bijection" title="Bijection">One-to-one correspondence</a></li> <li><a class="mw-selflink selflink">Ordinal number</a></li> <li><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li> <li><a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_set" title="Amorphous set">Amorphous</a></li> <li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a> (<a href="/wiki/Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li> <li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">Filter</a> <ul><li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">base</a></li> <li><a href="/wiki/Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li> <li><a href="/wiki/Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a> (<a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li> <li><a href="/wiki/Computable_set" title="Computable set">Recursive</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Subset" title="Subset">Subset <b>·</b> Superset</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li> <li><a href="/wiki/Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul> <ul><li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo</a> <ul><li><a href="/wiki/General_set_theory" title="General set theory">General</a></li></ul></li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> <ul><li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li></ul></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a> <ul><li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Set_theorists" title="Category:Set theorists">Set theorists</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li> <li><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link 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