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id="toc-17th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#17th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>17th century</span> </div> </a> <ul id="toc-17th_century-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Mathematics</span> </div> </a> <button aria-controls="toc-Mathematics-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 Mathematics subsection</span> </button> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> <li id="toc-Symbol" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symbol"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Symbol</span> </div> </a> <ul id="toc-Symbol-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Calculus</span> </div> </a> <ul id="toc-Calculus-sublist" class="vector-toc-list"> <li id="toc-Real_analysis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Real_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Real analysis</span> </div> </a> <ul id="toc-Real_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_analysis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Complex_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Complex analysis</span> </div> </a> <ul id="toc-Complex_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nonstandard_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonstandard_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Nonstandard analysis</span> </div> </a> <ul id="toc-Nonstandard_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Set_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Set theory</span> </div> </a> <ul id="toc-Set_theory-sublist" class="vector-toc-list"> <li id="toc-Cardinality_of_the_continuum" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cardinality_of_the_continuum"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.1</span> <span>Cardinality of the continuum</span> </div> </a> <ul id="toc-Cardinality_of_the_continuum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Geometry</span> </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_dimension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_dimension"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Infinite dimension</span> </div> </a> <ul id="toc-Infinite_dimension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Fractals</span> </div> </a> <ul id="toc-Fractals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematics_without_infinity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematics_without_infinity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Mathematics without infinity</span> </div> </a> <ul id="toc-Mathematics_without_infinity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Physics</span> </div> </a> <button aria-controls="toc-Physics-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 Physics subsection</span> </button> <ul id="toc-Physics-sublist" class="vector-toc-list"> <li id="toc-Cosmology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cosmology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Cosmology</span> </div> </a> <ul id="toc-Cosmology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Logic</span> </div> </a> <ul id="toc-Logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computing" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computing"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Computing</span> </div> </a> <ul id="toc-Computing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arts,_games,_and_cognitive_sciences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arts,_games,_and_cognitive_sciences"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Arts, games, and cognitive sciences</span> </div> </a> <ul id="toc-Arts,_games,_and_cognitive_sciences-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">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">8</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </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">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Infinity</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 115 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-115" 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">115 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Oneindigheid" title="Oneindigheid – Afrikaans" lang="af" hreflang="af" data-title="Oneindigheid" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Unendlichkeit" title="Unendlichkeit – Alemannic" lang="gsw" hreflang="gsw" data-title="Unendlichkeit" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%8B%95%E1%88%8B%E1%8D%8D" title="አዕላፍ – Amharic" lang="am" hreflang="am" data-title="አዕላፍ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D8%A7%D9%86%D9%87%D8%A7%D9%8A%D8%A9" title="لانهاية – Arabic" lang="ar" hreflang="ar" data-title="لانهاية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Infinito" title="Infinito – Aragonese" lang="an" hreflang="an" data-title="Infinito" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%85%E0%A6%B8%E0%A7%80%E0%A6%AE" title="অসীম – Assamese" lang="as" hreflang="as" data-title="অসীম" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Infinitu" title="Infinitu – Asturian" lang="ast" hreflang="ast" data-title="Infinitu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sonsuzluq" title="Sonsuzluq – Azerbaijani" lang="az" hreflang="az" data-title="Sonsuzluq" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%B3%D9%88%D9%86%D8%B3%D9%88%D8%B2" title="سونسوز – South Azerbaijani" lang="azb" hreflang="azb" data-title="سونسوز" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%B8%E0%A7%80%E0%A6%AE" title="অসীম – Bangla" lang="bn" hreflang="bn" data-title="অসীম" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Infinity" title="Infinity – Banjar" lang="bjn" hreflang="bjn" data-title="Infinity" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/B%C3%BB-h%C4%81n" title="Bû-hān – Minnan" lang="nan" hreflang="nan" data-title="Bû-hān" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BA%D2%BB%D0%B5%D2%99%D0%BB%D0%B5%D0%BA" title="Сикһеҙлек – Bashkir" lang="ba" hreflang="ba" data-title="Сикһеҙлек" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D0%B5%D1%81%D0%BA%D0%B0%D0%BD%D0%B5%D1%87%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Бесканечнасць – Belarusian" lang="be" hreflang="be" data-title="Бесканечнасць" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%91%D1%8F%D1%81%D0%BA%D0%BE%D0%BD%D1%86%D0%B0%D1%81%D1%8C%D1%86%D1%8C" title="Бясконцасьць – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Бясконцасьць" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B5%D0%B7%D0%BA%D1%80%D0%B0%D0%B9%D0%BD%D0%BE%D1%81%D1%82" title="Безкрайност – Bulgarian" lang="bg" hreflang="bg" data-title="Безкрайност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Beskona%C4%8Dnost" title="Beskonačnost – Bosnian" lang="bs" hreflang="bs" data-title="Beskonačnost" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Infinit" title="Infinit – Catalan" lang="ca" hreflang="ca" data-title="Infinit" 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%92%C4%95%C3%A7%D1%81%C4%95%D1%80%D0%BB%C4%95%D1%85" title="Вĕçсĕрлĕх – Chuvash" lang="cv" hreflang="cv" data-title="Вĕçсĕрлĕх" 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/Nekone%C4%8Dno" title="Nekonečno – Czech" lang="cs" hreflang="cs" data-title="Nekonečno" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Kusingaperi" title="Kusingaperi – Shona" lang="sn" hreflang="sn" data-title="Kusingaperi" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Infinitu" title="Infinitu – Corsican" lang="co" hreflang="co" data-title="Infinitu" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Anfeidredd" title="Anfeidredd – Welsh" lang="cy" hreflang="cy" data-title="Anfeidredd" 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/Uendelighed" title="Uendelighed – Danish" lang="da" hreflang="da" data-title="Uendelighed" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%84%D8%A7%D9%85%D8%B3%D8%A7%D9%84%D9%8A%D8%A9" title="لامسالية – Moroccan Arabic" lang="ary" hreflang="ary" data-title="لامسالية" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Unendlich_(Mathematik)" title="Unendlich (Mathematik) – German" lang="de" hreflang="de" data-title="Unendlich (Mathematik)" 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/L%C3%B5pmatus" title="Lõpmatus – Estonian" lang="et" hreflang="et" data-title="Lõpmatus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%86%CF%80%CE%B5%CE%B9%CF%81%CE%BF" title="Άπειρο – Greek" lang="el" hreflang="el" data-title="Άπειρο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Infinito" title="Infinito – Spanish" lang="es" hreflang="es" data-title="Infinito" 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/Senfineco" title="Senfineco – Esperanto" lang="eo" hreflang="eo" data-title="Senfineco" 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/Infinitu" title="Infinitu – Basque" lang="eu" hreflang="eu" data-title="Infinitu" 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%A8%DB%8C%E2%80%8C%D9%86%D9%87%D8%A7%DB%8C%D8%AA" title="بی‌نهایت – Persian" lang="fa" hreflang="fa" data-title="بی‌نهایت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Anant" title="Anant – Fiji Hindi" lang="hif" hreflang="hif" data-title="Anant" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Infini" title="Infini – French" lang="fr" hreflang="fr" data-title="Infini" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/%C3%89igr%C3%ADoch" title="Éigríoch – Irish" lang="ga" hreflang="ga" data-title="Éigríoch" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Infinito" title="Infinito – Galician" lang="gl" hreflang="gl" data-title="Infinito" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%84%A1%E9%99%90" title="無限 – Gan" lang="gan" hreflang="gan" data-title="無限" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%85%E0%AA%A8%E0%AA%82%E0%AA%A4" title="અનંત – Gujarati" lang="gu" hreflang="gu" data-title="અનંત" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AC%B4%ED%95%9C" title="무한 – Korean" lang="ko" hreflang="ko" data-title="무한" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%BE%D5%A5%D6%80%D5%BB%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Անվերջություն (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Անվերջություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A4%82%E0%A4%A4" title="अनंत – Hindi" lang="hi" hreflang="hi" data-title="अनंत" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Beskona%C4%8Dnost" title="Beskonačnost – Croatian" lang="hr" hreflang="hr" data-title="Beskonačnost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Infiniteso" title="Infiniteso – Ido" lang="io" hreflang="io" data-title="Infiniteso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Awan_inggana" title="Awan inggana – Iloko" lang="ilo" hreflang="ilo" data-title="Awan inggana" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Takhingga" title="Takhingga – Indonesian" lang="id" hreflang="id" data-title="Takhingga" 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/%C3%93endanleiki" title="Óendanleiki – Icelandic" lang="is" hreflang="is" data-title="Óendanleiki" 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/Infinito_(matematica)" title="Infinito (matematica) – Italian" lang="it" hreflang="it" data-title="Infinito (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%A0%D7%A1%D7%95%D7%A3" title="אינסוף – Hebrew" lang="he" hreflang="he" data-title="אינסוף" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%A8%E0%B2%82%E0%B2%A4" title="ಅನಂತ – Kannada" lang="kn" hreflang="kn" data-title="ಅನಂತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A3%E1%83%A1%E1%83%90%E1%83%A1%E1%83%A0%E1%83%A3%E1%83%9A%E1%83%9D%E1%83%91%E1%83%90" title="უსასრულობა – Georgian" lang="ka" hreflang="ka" data-title="უსასრულობა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B5%D0%BA%D1%81%D1%96%D0%B7%D0%B4%D1%96%D0%BA" title="Шексіздік – Kazakh" lang="kk" hreflang="kk" data-title="Шексіздік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Didhiwedhter" title="Didhiwedhter – Cornish" lang="kw" hreflang="kw" data-title="Didhiwedhter" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Enfini" title="Enfini – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Enfini" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/B%C3%AAdaw%C3%AE" title="Bêdawî – Kurdish" lang="ku" hreflang="ku" data-title="Bêdawî" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A7%D0%B5%D0%BA%D1%81%D0%B8%D0%B7%D0%B4%D0%B8%D0%BA" title="Чексиздик – Kyrgyz" lang="ky" hreflang="ky" data-title="Чексиздик" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://la.wikipedia.org/wiki/Infinitas" title="Infinitas – Latin" lang="la" hreflang="la" data-title="Infinitas" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bezgal%C4%ABba" title="Bezgalība – Latvian" lang="lv" hreflang="lv" data-title="Bezgalība" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Begalyb%C4%97" title="Begalybė – Lithuanian" lang="lt" hreflang="lt" data-title="Begalybė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/li_ci%27i" title="li ci'i – Lojban" lang="jbo" hreflang="jbo" data-title="li ci'i" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/V%C3%A9gtelen" title="Végtelen – Hungarian" lang="hu" hreflang="hu" data-title="Végtelen" 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%91%D0%B5%D1%81%D0%BA%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82" title="Бесконечност – Macedonian" lang="mk" hreflang="mk" data-title="Бесконечност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Tsiefa" title="Tsiefa – Malagasy" lang="mg" hreflang="mg" data-title="Tsiefa" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A8%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%A4" title="അനന്തത – Malayalam" lang="ml" hreflang="ml" data-title="അനന്തത" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A4%82%E0%A4%A4" title="अनंत – Marathi" lang="mr" hreflang="mr" data-title="अनंत" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D9%84%D9%87%D8%A7%D8%B4_%D9%86%D9%87%D8%A7%D9%8A%D9%87" title="ملهاش نهايه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="ملهاش نهايه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ketakterhinggaan" title="Ketakterhinggaan – Malay" lang="ms" hreflang="ms" data-title="Ketakterhinggaan" 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-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Tak_hingga" title="Tak hingga – Minangkabau" lang="min" hreflang="min" data-title="Tak hingga" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D1%8F%D0%B7%D0%B3%D0%B0%D0%B0%D1%80%D0%B3%D2%AF%D0%B9" title="Хязгааргүй – Mongolian" lang="mn" hreflang="mn" data-title="Хязгааргүй" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%94%E1%80%94%E1%80%B9%E1%80%90" title="အနန္တ – Burmese" lang="my" hreflang="my" data-title="အနန္တ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Oneindigheid" title="Oneindigheid – Dutch" lang="nl" hreflang="nl" data-title="Oneindigheid" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%84%A1%E9%99%90" title="無限 – Japanese" lang="ja" hreflang="ja" data-title="無限" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/%C3%9Cnentelkhaid" title="Ünentelkhaid – Northern Frisian" lang="frr" hreflang="frr" data-title="Ünentelkhaid" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Uendelig" title="Uendelig – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Uendelig" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Uendeleg" title="Uendeleg – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Uendeleg" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Infinit" title="Infinit – Occitan" lang="oc" hreflang="oc" data-title="Infinit" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Cheksizlik" title="Cheksizlik – Uzbek" lang="uz" hreflang="uz" data-title="Cheksizlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A8%A8%E0%A9%B0%E0%A8%A4" title="ਅਨੰਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਅਨੰਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%86%D8%A7%D9%86%D8%AA%DB%8C" title="انانتی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="انانتی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Infiniti" title="Infiniti – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Infiniti" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Infin%C3%AC" title="Infinì – Piedmontese" lang="pms" hreflang="pms" data-title="Infinì" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Unendlichkeid" title="Unendlichkeid – Low German" lang="nds" hreflang="nds" data-title="Unendlichkeid" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Niesko%C5%84czono%C5%9B%C4%87" title="Nieskończoność – Polish" lang="pl" hreflang="pl" data-title="Nieskończoność" 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/Infinito" title="Infinito – Portuguese" lang="pt" hreflang="pt" data-title="Infinito" 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/Infinit" title="Infinit – Romanian" lang="ro" hreflang="ro" data-title="Infinit" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%91%D0%B5%D1%81%D0%BA%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Бесконечность – Rusyn" lang="rue" hreflang="rue" data-title="Бесконечность" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B5%D1%81%D0%BA%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Бесконечность – Russian" lang="ru" hreflang="ru" data-title="Бесконечность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Infinity" title="Infinity – Scots" lang="sco" hreflang="sco" data-title="Infinity" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Pafund%C3%ABsia" title="Pafundësia – Albanian" lang="sq" hreflang="sq" data-title="Pafundësia" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Nfinitu_(matim%C3%A0tica)" title="Nfinitu (matimàtica) – Sicilian" lang="scn" hreflang="scn" data-title="Nfinitu (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%85%E0%B6%B1%E0%B6%B1%E0%B7%8A%E0%B6%AD%E0%B6%BA" title="අනන්තය – Sinhala" lang="si" hreflang="si" data-title="අනන්තය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Infinity" title="Infinity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Infinity" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Nekone%C4%8Dno" title="Nekonečno – Slovak" lang="sk" hreflang="sk" data-title="Nekonečno" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Neskon%C4%8Dnost" title="Neskončnost – Slovenian" lang="sl" hreflang="sl" data-title="Neskončnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%8E%DA%A9%DB%86%D8%AA%D8%A7%DB%8C%DB%8C" title="بێکۆتایی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بێکۆتایی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B5%D1%81%D0%BA%D0%BE%D0%BD%D0%B0%D1%87%D0%BD%D0%BE%D1%81%D1%82" title="Бесконачност – Serbian" lang="sr" hreflang="sr" data-title="Бесконачност" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Beskona%C4%8Dnost_(matematika)" title="Beskonačnost (matematika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Beskonačnost (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/%C3%84%C3%A4rett%C3%B6myys" title="Äärettömyys – Finnish" lang="fi" hreflang="fi" data-title="Äärettömyys" 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/O%C3%A4ndlighet" title="Oändlighet – Swedish" lang="sv" hreflang="sv" data-title="Oändlighet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Kawalang-hanggan" title="Kawalang-hanggan – Tagalog" lang="tl" hreflang="tl" data-title="Kawalang-hanggan" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%BF%E0%AE%B2%E0%AE%BF" title="முடிவிலி – Tamil" lang="ta" hreflang="ta" data-title="முடிவிலி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A7%D0%B8%D0%BA%D1%81%D0%B5%D0%B7%D0%BB%D0%B5%D0%BA" title="Чиксезлек – Tatar" lang="tt" hreflang="tt" data-title="Чиксезлек" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%99%E0%B8%B1%E0%B8%99%E0%B8%95%E0%B9%8C" title="อนันต์ – Thai" lang="th" hreflang="th" data-title="อนันต์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%91%D0%B5%D0%B8%D0%BD%D1%82%D0%B8%D2%B3%D0%BE%D3%A3" title="Беинтиҳоӣ – Tajik" lang="tg" hreflang="tg" data-title="Беинтиҳоӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sonsuz" title="Sonsuz – Turkish" lang="tr" hreflang="tr" data-title="Sonsuz" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B5%D1%81%D0%BA%D1%96%D0%BD%D1%87%D0%B5%D0%BD%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Нескінченність – Ukrainian" lang="uk" hreflang="uk" data-title="Нескінченність" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AA%D9%86%D8%A7%DB%81%DB%8C_%D8%A7%D9%88%D8%B1_%D9%84%D8%A7%D9%85%D8%AA%D9%86%D8%A7%DB%81%DB%8C" title="متناہی اور لامتناہی – Urdu" lang="ur" hreflang="ur" data-title="متناہی اور لامتناہی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Lopm%C3%A4tomuz" title="Lopmätomuz – Veps" lang="vep" hreflang="vep" data-title="Lopmätomuz" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%C3%B4_t%E1%BA%ADn" title="Vô tận – Vietnamese" lang="vi" hreflang="vi" data-title="Vô tận" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%84%A1%E9%99%90" title="無限 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="無限" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Infinidad" title="Infinidad – Waray" lang="war" hreflang="war" data-title="Infinidad" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%97%A0%E7%A9%B7" title="无穷 – Wu" lang="wuu" hreflang="wuu" data-title="无穷" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%95%D7%9E%D7%A2%D7%A0%D7%93%D7%9C%D7%A2%D7%9B%D7%A7%D7%99%D7%99%D7%98" title="אומענדלעכקייט – Yiddish" lang="yi" hreflang="yi" data-title="אומענדלעכקייט" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%84%A1%E7%AA%AE%E7%9B%A1" title="無窮盡 – Cantonese" lang="yue" hreflang="yue" data-title="無窮盡" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Begal%C4%ABb%C4%97" 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.hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable"><span>For the symbol, see <a href="/wiki/Infinity_symbol" title="Infinity symbol">Infinity symbol</a>. For other uses of "Infinity" and "Infinite", see <a href="/wiki/Infinity_(disambiguation)" class="mw-disambig" title="Infinity (disambiguation)">Infinity (disambiguation)</a>.</span> <span>Not to be confused with <a href="/wiki/Infiniti" title="Infiniti">Infiniti</a>.</span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SierpinskiTriangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/SierpinskiTriangle.svg/250px-SierpinskiTriangle.svg.png" decoding="async" width="220" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/SierpinskiTriangle.svg/330px-SierpinskiTriangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/SierpinskiTriangle.svg/500px-SierpinskiTriangle.svg.png 2x" data-file-width="744" data-file-height="644" /></a><figcaption>The <a href="/wiki/Sierpi%C5%84ski_triangle" title="Sierpiński triangle">Sierpiński triangle</a> contains infinitely many (scaled-down) copies of itself.</figcaption></figure> <p><b>Infinity</b> is something which is boundless, endless, or larger than any <a href="/wiki/Natural_number" title="Natural number">natural number</a>. It is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /></span>, the <a href="/wiki/Infinity_symbol" title="Infinity symbol">infinity symbol</a>. </p><p>From the time of the <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greeks</a>, the <a href="/wiki/Infinity_(philosophy)" title="Infinity (philosophy)">philosophical nature of infinity</a> has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol<sup id="cite_ref-:1_1-0" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a>, mathematicians began to work with <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> and what some mathematicians (including <a href="/wiki/Guillaume_de_l%27H%C3%B4pital" title="Guillaume de l'Hôpital">l'Hôpital</a> and <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Bernoulli</a>)<sup id="cite_ref-Jesseph_2-0" class="reference"><a href="#cite_note-Jesseph-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of <a href="/wiki/Calculus" title="Calculus">calculus</a>, it remained unclear whether infinity could be considered as a number or <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> and, if so, how this could be done.<sup id="cite_ref-:1_1-1" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> At the end of the 19th century, <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> enlarged the mathematical study of infinity by studying <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a> and <a href="/wiki/Transfinite_number" title="Transfinite number">infinite numbers</a>, showing that they can be of various sizes.<sup id="cite_ref-:1_1-2" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the line) is larger than the number of <a href="/wiki/Integer" title="Integer">integers</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> In this usage, infinity is a mathematical concept, and infinite <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> can be studied, manipulated, and used just like any other mathematical object. </p><p>The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>, on which most of modern mathematics can be developed, is the <a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">axiom of infinity</a>, which guarantees the existence of infinite sets.<sup id="cite_ref-:1_1-3" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> that may seem to have nothing to do with them. For example, <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles's proof of Fermat's Last Theorem</a> implicitly relies on the existence of <a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck universes</a>, very large infinite sets,<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> for solving a long-standing problem that is stated in terms of <a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a>. </p><p>In <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Cosmology" title="Cosmology">cosmology</a>, it is an open question <a href="/wiki/Universe#Size_and_regions" title="Universe">whether the universe is spatially infinite or not</a>. </p> <meta property="mw:PageProp/toc" /> <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=Infinity&action=edit&section=1" title="Edit section: History"><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/Infinity_(philosophy)" title="Infinity (philosophy)">Infinity (philosophy)</a></div> <p>Ancient cultures had various ideas about the nature of infinity. The <a href="/wiki/Vedic_period" title="Vedic period">ancient Indians</a> and the <a href="/wiki/Ancient_Greece" title="Ancient Greece">Greeks</a> did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. </p> <div class="mw-heading mw-heading3"><h3 id="Early_Greek">Early Greek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=2" title="Edit section: Early Greek"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The earliest recorded idea of infinity in Greece may be that of <a href="/wiki/Anaximander" title="Anaximander">Anaximander</a> (c. 610 – c. 546 BC) a <a href="/wiki/Pre-Socratic_philosophy" title="Pre-Socratic philosophy">pre-Socratic</a> Greek philosopher. He used the word <i><a href="/wiki/Apeiron" title="Apeiron">apeiron</a></i>, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".<sup id="cite_ref-:1_1-4" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (350 BC) distinguished <i>potential infinity</i> from <i><a href="/wiki/Actual_infinity" title="Actual infinity">actual infinity</a></i>, which he regarded as impossible due to the various <a href="/wiki/Paradoxes" class="mw-redirect" title="Paradoxes">paradoxes</a> it seemed to produce.<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> It has been argued that, in line with this view, the <a href="/wiki/Hellenistic" class="mw-redirect" title="Hellenistic">Hellenistic</a> Greeks had a "horror of the infinite"<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><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> which would, for example, explain why <a href="/wiki/Euclid" title="Euclid">Euclid</a> (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers."<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> It has also been maintained, that, in proving the <a href="/wiki/Infinitude_of_the_prime_numbers" class="mw-redirect" title="Infinitude of the prime numbers">infinitude of the prime numbers</a>, Euclid "was the first to overcome the horror of the infinite".<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> There is a similar controversy concerning Euclid's <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>, sometimes translated: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.<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></blockquote> <p>Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...",<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> thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.<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> <div class="mw-heading mw-heading3"><h3 id="Zeno:_Achilles_and_the_tortoise">Zeno: Achilles and the tortoise</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=3" title="Edit section: Zeno: Achilles and the tortoise"><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/Zeno%27s_paradoxes#Achilles_and_the_tortoise" title="Zeno's paradoxes">Zeno's paradoxes § Achilles and the tortoise</a></div> <p><a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a> (<abbr title="circa">c.</abbr> 495 – <abbr title="circa">c.</abbr> 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,<sup id="cite_ref-Zeno's_paradoxes_15-0" class="reference"><a href="#cite_note-Zeno's_paradoxes-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> as "immeasurably subtle and profound".<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> </p><p><a href="/wiki/Achilles" title="Achilles">Achilles</a> races a tortoise, giving the latter a head start. </p> <ul><li>Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.</li> <li>Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.</li> <li>Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.</li> <li>Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.</li></ul> <p>Etc. </p><p>Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him. </p><p>Zeno was not attempting to make a point about infinity. As a member of the <a href="/wiki/Eleatic" class="mw-redirect" title="Eleatic">Eleatics</a> school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument. </p><p>Finally, in 1821, <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> provided both a satisfactory definition of a limit and a proof that, for <span class="texhtml">0 < <i>x</i> < 1</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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/273688d4c215ab7a8c65ccbe30c52a678b550c0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.924ex; height:4.843ex;" alt="{\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}" /></span> </p><p>Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with <span class="texhtml"><i>a</i> = 10 seconds</span> and <span class="texhtml"><i>x</i> = 0.01</span>. Achilles does overtake the tortoise; it takes him </p> <dl><dd><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 10+0.1+0.001+0.00001+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mo>+</mo> <mn>0.1</mn> <mo>+</mo> <mn>0.001</mn> <mo>+</mo> <mn>0.00001</mn> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10+0.1+0.001+0.00001+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c12168beb594444f2ce8de7636bd767228f97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:32.3ex; height:2.343ex;" alt="{\displaystyle 10+0.1+0.001+0.00001+\cdots }" /></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 ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mrow> <mn>1</mn> <mo>−</mo> <mn>0.01</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>0.99</mn> </mfrac> </mrow> <mo>=</mo> <mn>10.10101</mn> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> seconds</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26c77fb424926d2688b784ff26fb51f35cfce25b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.748ex; height:5.343ex;" alt="{\displaystyle ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Early_Indian">Early Indian</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=4" title="Edit section: Early Indian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Indian_mathematics" title="Indian mathematics">Jain mathematical</a> text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: <a href="/wiki/Enumerable" class="mw-redirect" title="Enumerable">enumerable</a>, innumerable, and infinite. Each of these was further subdivided into three orders:<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> </p> <ul><li>Enumerable: lowest, intermediate, and highest</li> <li>Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable</li> <li>Infinite: nearly infinite, truly infinite, infinitely infinite</li></ul> <div class="mw-heading mw-heading3"><h3 id="17th_century">17th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=5" title="Edit section: 17th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> first used 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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /></span> for such a number in his <i>De sectionibus conicis</i>,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and exploited it in area calculations by dividing the region into <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> strips of width on the order 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 {\tfrac {1}{\infty }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi mathvariant="normal">∞</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\infty }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/286a6d82b16111d4ae231286d8e66bb0c9713d7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.126ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{\infty }}.}" /></span><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> But in <i>Arithmetica infinitorum</i> (1656),<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>In 1699, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> wrote about equations with an infinite number of terms in his work <i><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De analysi per aequationes numero terminorum infinitas</a></i>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=6" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> opened a mathematico-philosophic address given in 1930 with:<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote"><p>Mathematics is the science of the infinite.</p></blockquote> <div class="mw-heading mw-heading3"><h3 id="Symbol">Symbol</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=7" title="Edit section: Symbol"><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/Infinity_symbol" title="Infinity symbol">Infinity symbol</a></div> <p>The infinity symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /></span> (sometimes called the <a href="/wiki/Lemniscate" title="Lemniscate">lemniscate</a>) is a mathematical symbol representing the concept of infinity. The symbol is encoded in <a href="/wiki/Unicode" title="Unicode">Unicode</a> at <span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+221E</span> </span><span style="font-size:125%;line-height:1em">∞</span> <span style="font-variant: small-caps; text-transform: lowercase; font-feature-settings: 'onum'">INFINITY</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">&infin;</span>)<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and in <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> as <code>\infty</code>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>It was introduced in 1655 by <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a>,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> and since its introduction, it has also been used outside mathematics in modern mysticism<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> and literary <a href="/wiki/Symbology" class="mw-redirect" title="Symbology">symbology</a>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Calculus">Calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=8" title="Edit section: Calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Leibniz</a>, one of the co-inventors of <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a>, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the <a href="/wiki/Law_of_continuity" title="Law of continuity">Law of continuity</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Jesseph_2-1" class="reference"><a href="#cite_note-Jesseph-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Real_analysis">Real analysis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=9" title="Edit section: Real analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /></span>, called "infinity", is used to denote an unbounded <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> 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 x\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02579e74e2ef1ca0befceba816b311fe5bfd6844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\rightarrow \infty }" /></span> means that <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span></i> increases without bound, 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 x\to -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a02931d12f93c144745ec549edf61e85fba2c3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.076ex; height:2.176ex;" alt="{\displaystyle x\to -\infty }" /></span> means that <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span></i> decreases without bound. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f73da3eb2fb3bf9f777a76c755f66313b3a9061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.188ex; height:2.843ex;" alt="{\displaystyle f(t)\geq 0}" /></span> for every <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span></i>, then<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ac86f8a3cde818f839059bf58d6c73900a06ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.581ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }" /></span> means 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 f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" /></span> does not bound a finite area from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span> 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 b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef051eb30c89e5493d672f6479566c673b0890a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.644ex; height:2.176ex;" alt="{\displaystyle b.}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6c7b696e4d5e2cb178d5b448f0c7e599eb7238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.626ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }" /></span> means that the area 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 f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" /></span> is infinite.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/342e8f515d11881d59a0abf355455bada0d6e9e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.532ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}" /></span> means that the total area 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 f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" /></span> is finite, and is 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 a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b803da9c45c1186883bde55107e9ccb102c92c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}" /></span></li></ul> <p>Infinity can also be used to describe <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>, as follows: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{\infty }f(i)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{\infty }f(i)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0539deb728204aadf32840ba0e72b8ec3ba8cc96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.961ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{\infty }f(i)=a}" /></span> means that the sum of the infinite series <a href="/wiki/Convergent_series" title="Convergent series">converges</a> to some real value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b803da9c45c1186883bde55107e9ccb102c92c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11178ed4dc89a3d6a502221d42b91bac4779d545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.055ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }" /></span> means that the sum of the infinite series properly <a href="/wiki/Divergent_series" title="Divergent series">diverges</a> to infinity, in the sense that the partial sums increase without bound.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup></li></ul> <p>In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled <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 +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" /></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 -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" /></span> can be added to the <a href="/wiki/Topological_space" title="Topological space">topological space</a> of the real numbers, producing the two-point <a href="/wiki/Compactification_(mathematics)" title="Compactification (mathematics)">compactification</a> of the real numbers. Adding algebraic properties to this gives us the <a href="/wiki/Extended_real_number" class="mw-redirect" title="Extended real number">extended real numbers</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> We can also treat <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 +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" /></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 -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" /></span> as the same, leading to the <a href="/wiki/One-point_compactification" class="mw-redirect" title="One-point compactification">one-point compactification</a> of the real numbers, which is the <a href="/wiki/Real_projective_line" title="Real projective line">real projective line</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Projective_geometry" title="Projective geometry">Projective geometry</a> also refers to a <a href="/wiki/Line_at_infinity" title="Line at infinity">line at infinity</a> in plane geometry, a <a href="/wiki/Plane_at_infinity" title="Plane at infinity">plane at infinity</a> in three-dimensional space, and a <a href="/wiki/Hyperplane_at_infinity" title="Hyperplane at infinity">hyperplane at infinity</a> for general <a href="/wiki/Dimension_(mathematics_and_physics)" class="mw-redirect" title="Dimension (mathematics and physics)">dimensions</a>, each consisting of <a href="/wiki/Point_at_infinity" title="Point at infinity">points at infinity</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Complex_analysis">Complex analysis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=10" title="Edit section: Complex analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Riemann_sphere1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Riemann_sphere1.svg/250px-Riemann_sphere1.svg.png" decoding="async" width="250" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Riemann_sphere1.svg/375px-Riemann_sphere1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Riemann_sphere1.svg/500px-Riemann_sphere1.svg.png 2x" data-file-width="800" data-file-height="640" /></a><figcaption>By <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>.</figcaption></figure> <p>In <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /></span>, called "infinity", denotes an unsigned infinite <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>. The expression <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02579e74e2ef1ca0befceba816b311fe5bfd6844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\rightarrow \infty }" /></span> means that the magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb41e5fd5dc37eaa1718dfbf4bc082edb991936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle |x|}" /></span> of <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span></i> grows beyond any assigned value. A <a href="/wiki/Point_at_infinity" title="Point at infinity">point labeled <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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /></span></a> can be added to the complex plane as a <a href="/wiki/Topological_space" title="Topological space">topological space</a> giving the <a href="/wiki/One-point_compactification" class="mw-redirect" title="One-point compactification">one-point compactification</a> of the complex plane. When this is done, the resulting space is a one-dimensional <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a>, or <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a>, called the extended complex plane or the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a>, namely <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 z/0=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>0</mn> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z/0=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8796ee3f88b038e113bfd7a20633183722a6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.835ex; height:2.843ex;" alt="{\displaystyle z/0=\infty }" /></span> for any nonzero <a href="/wiki/Complex_number" title="Complex number">complex number</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span></i>. In this context, it is often useful to consider <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic functions</a> as maps into the Riemann sphere taking the value 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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /></span> at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> (see <a href="/wiki/M%C3%B6bius_transformation#Overview" title="Möbius transformation">Möbius transformation § Overview</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Nonstandard_analysis">Nonstandard analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=11" title="Edit section: Nonstandard analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:N%C3%BAmeros_hiperreales.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/N%C3%BAmeros_hiperreales.png/500px-N%C3%BAmeros_hiperreales.png" decoding="async" width="450" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/N%C3%BAmeros_hiperreales.png/675px-N%C3%BAmeros_hiperreales.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/53/N%C3%BAmeros_hiperreales.png 2x" data-file-width="804" data-file-height="297" /></a><figcaption>Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)</figcaption></figure> <p>The original formulation of <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a> by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various <a href="/wiki/Logical_system" class="mw-redirect" title="Logical system">logical systems</a>, including <a href="/wiki/Smooth_infinitesimal_analysis" title="Smooth infinitesimal analysis">smooth infinitesimal analysis</a> and <a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">nonstandard analysis</a>. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal field</a>; there is no equivalence between them as with the Cantorian <a href="/wiki/Transfinite_number" title="Transfinite number">transfinites</a>. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to <a href="/wiki/Non-standard_calculus" class="mw-redirect" title="Non-standard calculus">non-standard calculus</a> is fully developed in <a href="#CITEREFKeisler1986">Keisler (1986)</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Set_theory">Set theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=12" title="Edit section: Set theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Cardinality" title="Cardinality">Cardinality</a> and <a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif/250px-Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif" decoding="async" width="220" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif/330px-Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/a/ab/Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif 2x" data-file-width="424" data-file-height="221" /></a><figcaption>One-to-one correspondence between an infinite set and its proper subset</figcaption></figure> <p>A different form of "infinity" is the <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> and <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal</a> infinities of set theory—a system of <a href="/wiki/Transfinite_number" title="Transfinite number">transfinite numbers</a> first developed by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>. In this system, the first transfinite cardinal is <a href="/wiki/Aleph-null" class="mw-redirect" title="Aleph-null">aleph-null</a> (<span style="font-family:'Cambria Math';"><big>ℵ</big><sub>0</sub></span>), the cardinality of the set of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>, <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> and others—using the idea of collections or sets.<sup id="cite_ref-:1_1-5" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Dedekind's approach was essentially to adopt the idea of <a href="/wiki/One-to-one_correspondence" class="mw-redirect" title="One-to-one correspondence">one-to-one correspondence</a> as a standard for comparing the size of sets, and to reject the view of Galileo (derived from <a href="/wiki/Euclid" title="Euclid">Euclid</a>) that the whole cannot be the same size as the part. (However, see <a href="/wiki/Galileo%27s_paradox" title="Galileo's paradox">Galileo's paradox</a> where Galileo concludes that positive integers cannot be compared to the subset of positive <a href="/wiki/Square_number" title="Square number">square integers</a> since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper</a> parts; this notion of infinity is called <a href="/wiki/Dedekind_infinite" class="mw-redirect" title="Dedekind infinite">Dedekind infinite</a>. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>Cantor defined two kinds of infinite numbers: <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a> and <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>. Ordinal numbers characterize <a href="/wiki/Well-ordered" class="mw-redirect" title="Well-ordered">well-ordered</a> sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite <a href="/wiki/Sequence" title="Sequence">sequences</a> which are maps from the positive <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> leads to <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">mappings</a> from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is <a href="/wiki/Countable_set" title="Countable set">countably infinite</a>. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called <i><a href="/wiki/Uncountable_set" title="Uncountable set">uncountable</a></i>. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Certain extended number systems, such as the <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal numbers</a>, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Cardinality_of_the_continuum">Cardinality of the continuum</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=13" title="Edit section: Cardinality of the continuum"><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/Cardinality_of_the_continuum" title="Cardinality of the continuum">Cardinality of the continuum</a></div><p> One of Cantor's most important results was that the cardinality of the continuum <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 \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8798d172f59e21f2ce193a3118d4063d19353ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {c} }" /></span> is greater than that of the natural numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\aleph _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </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/98d41341ae020c937dee3df105f059eb93ed96c8" 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>; that is, there are more real numbers <span class="texhtml"><b>R</b></span> than natural numbers <span class="texhtml"><b>N</b></span>. Namely, Cantor showed 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 \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d9e38e6111127ba395d142775504a2dbe9fe55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.09ex; height:3.009ex;" alt="{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}" /></span>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /></p><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Cantor's diagonal argument</a> and <a href="/wiki/Cantor%27s_first_set_theory_article" title="Cantor's first set theory article">Cantor's first set theory article</a></div><p> The <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> states that there is no <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a> between the cardinality of the reals and the cardinality of the natural numbers, that 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 \mathbf {c} =\aleph _{1}=\beth _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>ℶ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ad512ea947a600dace4fec2372d2f9ef1f6c20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.464ex; height:2.676ex;" alt="{\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}" /></span>.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /></p><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Beth_number#Beth_one" title="Beth number">Beth number § Beth one</a></div><p>This hypothesis cannot be proved or disproved within the widely accepted <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>, even assuming the <a href="/wiki/Axiom_of_Choice" class="mw-redirect" title="Axiom of Choice">Axiom of Choice</a>.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Cardinal_arithmetic" class="mw-redirect" title="Cardinal arithmetic">Cardinal arithmetic</a> can be used to show not only that the number of points in a <a href="/wiki/Real_number_line" class="mw-redirect" title="Real number line">real number line</a> is equal to the number of points in any <a href="/wiki/Line_segment" title="Line segment">segment of that line</a>, but also that this is equal to the number of points on a plane and, indeed, in any <a href="/wiki/Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a> space.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Peanocurve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Peanocurve.svg/250px-Peanocurve.svg.png" decoding="async" width="220" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Peanocurve.svg/330px-Peanocurve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Peanocurve.svg/500px-Peanocurve.svg.png 2x" data-file-width="930" data-file-height="284" /></a><figcaption>The first three steps of a fractal construction whose limit is a <a href="/wiki/Space-filling_curve" title="Space-filling curve">space-filling curve</a>, showing that there are as many points in a one-dimensional line as in a two-dimensional square</figcaption></figure><p> The first of these results is apparent by considering, for instance, the <a href="/wiki/Tangent_(trigonometric_function)" class="mw-redirect" title="Tangent (trigonometric function)">tangent</a> function, which provides a <a href="/wiki/One-to-one_correspondence" class="mw-redirect" title="One-to-one correspondence">one-to-one correspondence</a> between the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> (<span class="texhtml">−<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>) and<span class="texhtml"> <b>R</b></span>.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /></p><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's paradox of the Grand Hotel</a></div><p>The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> introduced the <a href="/wiki/Space-filling_curve" title="Space-filling curve">space-filling curves</a>, curved lines that twist and turn enough to fill the whole of any square, or <a href="/wiki/Cube" title="Cube">cube</a>, or <a href="/wiki/Hypercube" title="Hypercube">hypercube</a>, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><div class="mw-heading mw-heading3"><h3 id="Geometry">Geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=14" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Until the end of the 19th century, infinity was rarely discussed in <a href="/wiki/Geometry" title="Geometry">geometry</a>, except in the context of processes that could be continued without any limit. For example, a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> was what is now called a <a href="/wiki/Line_segment" title="Line segment">line segment</a>, with the proviso that one can extend it as far as one wants; but extending it <i>infinitely</i> was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of <i>a point</i> that satisfies some property" (singular), where modern mathematicians would generally say "the set of <i>the points</i> that have the property" (plural). </p><p>One of the rare exceptions of a mathematical concept involving <a href="/wiki/Actual_infinity" title="Actual infinity">actual infinity</a> was <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, where <a href="/wiki/Points_at_infinity" class="mw-redirect" title="Points at infinity">points at infinity</a> are added to the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> for modeling the <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a> effect that shows <a href="/wiki/Parallel_lines" class="mw-redirect" title="Parallel lines">parallel lines</a> intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a>, two distinct <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a> intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry. </p><p>Before the use of <a href="/wiki/Set_theory" title="Set theory">set theory</a> for the <a href="/wiki/Foundation_of_mathematics" class="mw-redirect" title="Foundation of mathematics">foundation of mathematics</a>, points and lines were viewed as distinct entities, and a point could be <i>located on a line</i>. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as <i>the set of its points</i>, and one says that a point <i>belongs to a line</i> instead of <i>is located on a line</i> (however, the latter phrase is still used). </p><p>In particular, in modern mathematics, lines are <i>infinite sets</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Infinite_dimension">Infinite dimension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=15" title="Edit section: Infinite dimension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> that occur in classical <a href="/wiki/Geometry" title="Geometry">geometry</a> have always a finite <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a>, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> where <a href="/wiki/Function_space" title="Function space">function spaces</a> are generally vector spaces of infinite dimension. </p><p>In topology, some constructions can generate <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> of infinite dimension. In particular, this is the case of <a href="/wiki/Iterated_loop_space" class="mw-redirect" title="Iterated loop space">iterated loop spaces</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Fractals">Fractals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=16" title="Edit section: Fractals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The structure of a <a href="/wiki/Fractal" title="Fractal">fractal</a> object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters and can have infinite or finite areas. One such <a href="/wiki/Fractal_curve" title="Fractal curve">fractal curve</a> with an infinite perimeter and finite area is the <a href="/wiki/Koch_snowflake" title="Koch snowflake">Koch snowflake</a>.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Mathematics_without_infinity">Mathematics without infinity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=17" title="Edit section: Mathematics without infinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a> was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a> called <a href="/wiki/Finitism" title="Finitism">finitism</a>, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of <a href="/wiki/Mathematical_constructivism" class="mw-redirect" title="Mathematical constructivism">constructivism</a> and <a href="/wiki/Intuitionism" title="Intuitionism">intuitionism</a>.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Physics">Physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=18" title="Edit section: Physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, approximations of <a href="/wiki/Real_number" title="Real number">real numbers</a> are used for <a href="/wiki/Continuum_(theory)" class="mw-redirect" title="Continuum (theory)">continuous</a> measurements and <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> are used for <a href="/wiki/Countable" class="mw-redirect" title="Countable">discrete</a> measurements (i.e., counting). Concepts of infinite things such as an infinite <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a> exist, but there are no experimental means to generate them.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Cosmology">Cosmology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=19" title="Edit section: Cosmology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first published proposal that the universe is infinite came from Thomas Digges in 1576.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Eight years later, in 1584, the Italian philosopher and astronomer <a href="/wiki/Giordano_Bruno" title="Giordano Bruno">Giordano Bruno</a> proposed an unbounded universe in <i>On the Infinite Universe and Worlds</i>: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Cosmology" title="Cosmology">Cosmologists</a> have long sought to discover whether infinity exists in our physical <a href="/wiki/Universe" title="Universe">universe</a>: Are there an infinite number of stars? Does the universe have infinite volume? Does space "<a href="/wiki/Shape_of_the_universe" title="Shape of the universe">go on forever</a>"? This is still an open question of <a href="/wiki/Physical_cosmology" title="Physical cosmology">cosmology</a>. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar <a href="/wiki/Topology" title="Topology">topology</a>. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p>The curvature of the universe can be measured through <a href="/wiki/Multipole_moments" class="mw-redirect" title="Multipole moments">multipole moments</a> in the spectrum of the <a href="/wiki/Cosmic_microwave_background_radiation" class="mw-redirect" title="Cosmic microwave background radiation">cosmic background radiation</a>. To date, analysis of the radiation patterns recorded by the <a href="/wiki/WMAP" class="mw-redirect" title="WMAP">WMAP</a> spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.<sup id="cite_ref-NASA_Shape_52-0" class="reference"><a href="#cite_note-NASA_Shape-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Fermi_Flat_53-0" class="reference"><a href="#cite_note-Fermi_Flat-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p><p>However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is <a href="/wiki/Torus" title="Torus">toroidal</a> and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>The concept of infinity also extends to the <a href="/wiki/Multiverse" title="Multiverse">multiverse</a> hypothesis, which, when explained by astrophysicists such as <a href="/wiki/Michio_Kaku" title="Michio Kaku">Michio Kaku</a>, posits that there are an infinite number and variety of universes.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> Also, <a href="/wiki/Cyclic_model" title="Cyclic model">cyclic models</a> posit an infinite amount of <a href="/wiki/Big_Bang" title="Big Bang">Big Bangs</a>, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.<sup id="cite_ref-Nautilus2014_57-0" class="reference"><a href="#cite_note-Nautilus2014-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Logic">Logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=20" title="Edit section: Logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Logic" title="Logic">logic</a>, an <a href="/wiki/Infinite_regress" title="Infinite regress">infinite regress</a> argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Computing">Computing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=21" title="Edit section: Computing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/IEEE_floating-point" class="mw-redirect" title="IEEE floating-point">IEEE floating-point</a> standard (IEEE 754) specifies a positive and a negative infinity value (and also <a href="/wiki/NaN" title="NaN">indefinite</a> values). These are defined as the result of <a href="/wiki/Arithmetic_overflow" class="mw-redirect" title="Arithmetic overflow">arithmetic overflow</a>, <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a>, and other exceptional operations.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>Some <a href="/wiki/Programming_language" title="Programming language">programming languages</a>, such as <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/J_(programming_language)" title="J (programming language)">J</a>,<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as <a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest and least elements</a>, as they compare (respectively) greater than or less than all other values. They have uses as <a href="/wiki/Sentinel_value" title="Sentinel value">sentinel values</a> in <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> involving <a href="/wiki/Sorting" title="Sorting">sorting</a>, <a href="/wiki/Search_algorithm" title="Search algorithm">searching</a>, or <a href="/wiki/Window_function" title="Window function">windowing</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2017)">citation needed</span></a></i>]</sup> </p><p>In languages that do not have greatest and least elements but do allow <a href="/wiki/Operator_overloading" title="Operator overloading">overloading</a> of <a href="/wiki/Relational_operator" title="Relational operator">relational operators</a>, it is possible for a programmer to <i>create</i> the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program but do implement the floating-point <a href="/wiki/Data_type" title="Data type">data type</a>, the infinity values may still be accessible and usable as the result of certain operations.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2017)">citation needed</span></a></i>]</sup> </p><p>In programming, an <a href="/wiki/Infinite_loop" title="Infinite loop">infinite loop</a> is a <a href="/wiki/Loop_(computing)" class="mw-redirect" title="Loop (computing)">loop</a> whose exit condition is never satisfied, thus executing indefinitely. </p> <div class="mw-heading mw-heading2"><h2 id="Arts,_games,_and_cognitive_sciences"><span id="Arts.2C_games.2C_and_cognitive_sciences"></span>Arts, games, and cognitive sciences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=22" title="Edit section: Arts, games, and cognitive sciences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">Perspective</a> artwork uses the concept of <a href="/wiki/Vanishing_point" title="Vanishing point">vanishing points</a>, roughly corresponding to mathematical <a href="/wiki/Point_at_infinity" title="Point at infinity">points at infinity</a>, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> Artist <a href="/wiki/M.C._Escher" class="mw-redirect" title="M.C. Escher">M.C. Escher</a> is specifically known for employing the concept of infinity in his work in this and other ways.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p><p>Variations of <a href="/wiki/Chess" title="Chess">chess</a> played on an unbounded board are called <a href="/wiki/Infinite_chess" title="Infinite chess">infinite chess</a>.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Cognitive_science" title="Cognitive science">Cognitive scientist</a> <a href="/wiki/George_Lakoff" title="George Lakoff">George Lakoff</a> considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1, 2, 3, …>.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </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=Infinity&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/0.999..." title="0.999...">0.999...</a></li> <li><a href="/wiki/Absolute_infinite" title="Absolute infinite">Absolute infinite</a></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Ananta_(infinite)" title="Ananta (infinite)">Ananta</a></li> <li><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a></li> <li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Names_of_large_numbers" title="Names of large numbers">Names of large numbers</a></li> <li><a href="/wiki/Infinite_monkey_theorem" title="Infinite monkey theorem">Infinite monkey theorem</a></li> <li><a href="/wiki/Paradoxes_of_infinity" class="mw-redirect" title="Paradoxes of infinity">Paradoxes of infinity</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal number</a></li></ul> </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=Infinity&action=edit&section=24" title="Edit section: References"><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-:1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:1_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:1_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:1_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAllen2003" class="citation web cs1">Allen, Donald (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200801202539/https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf">"The History of Infinity"</a> <span class="cs1-format">(PDF)</span>. <i>Texas A&M Mathematics</i>. Archived from <a rel="nofollow" class="external text" href="https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf">the original</a> <span class="cs1-format">(PDF)</span> on August 1, 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">Nov 15,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Texas+A%26M+Mathematics&rft.atitle=The+History+of+Infinity&rft.date=2003&rft.aulast=Allen&rft.aufirst=Donald&rft_id=https%3A%2F%2Fwww.math.tamu.edu%2F~dallen%2Fmasters%2Finfinity%2Finfinity.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-Jesseph-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Jesseph_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Jesseph_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJesseph1998" class="citation journal cs1">Jesseph, Douglas Michael (1998-05-01). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120111102635/http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html">"Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes"</a>. <i><a href="/wiki/Perspectives_on_Science" title="Perspectives on Science">Perspectives on Science</a></i>. <b>6</b> (1&2): <span class="nowrap">6–</span>40. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Fposc_a_00543">10.1162/posc_a_00543</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1063-6145">1063-6145</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/42413222">42413222</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:118227996">118227996</a>. Archived from <a rel="nofollow" class="external text" href="http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html">the original</a> on 11 January 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">1 November</span> 2019</span> – via Project MUSE.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Perspectives+on+Science&rft.atitle=Leibniz+on+the+Foundations+of+the+Calculus%3A+The+Question+of+the+Reality+of+Infinitesimal+Magnitudes&rft.volume=6&rft.issue=1%262&rft.pages=%3Cspan+class%3D%22nowrap%22%3E6-%3C%2Fspan%3E40&rft.date=1998-05-01&rft_id=info%3Aoclcnum%2F42413222&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118227996%23id-name%3DS2CID&rft.issn=1063-6145&rft_id=info%3Adoi%2F10.1162%2Fposc_a_00543&rft.aulast=Jesseph&rft.aufirst=Douglas+Michael&rft_id=http%3A%2F%2Fmuse.jhu.edu%2Fjournals%2Fperspectives_on_science%2Fv006%2F6.1jesseph.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGowersBarrow-Green2008" class="citation book cs1">Gowers, Timothy; Barrow-Green, June (2008). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/659590835"><i>The Princeton companion to mathematics</i></a>. Imre Leader, Princeton University. Princeton: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-3039-8" title="Special:BookSources/978-1-4008-3039-8"><bdi>978-1-4008-3039-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/659590835">659590835</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Princeton+companion+to+mathematics&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=2008&rft_id=info%3Aoclcnum%2F659590835&rft.isbn=978-1-4008-3039-8&rft.aulast=Gowers&rft.aufirst=Timothy&rft.au=Barrow-Green%2C+June&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F659590835&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFMaddox2002">Maddox 2002</a>, pp. 113–117</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="CITEREFMcLarty2014" class="citation journal cs1">McLarty, Colin (15 January 2014) [September 2010]. <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/abs/what-does-it-take-to-prove-fermats-last-theorem-grothendieck-and-the-logic-of-number-theory/80EDFF3616F8D58590EBA0DCB9FD2E3E">"What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory"</a>. <i>The Bulletin of Symbolic Logic</i>. <b>16</b> (3): <span class="nowrap">359–</span>377. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2178%2Fbsl%2F1286284558">10.2178/bsl/1286284558</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:13475845">13475845</a> – via Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Bulletin+of+Symbolic+Logic&rft.atitle=What+Does+it+Take+to+Prove+Fermat%27s+Last+Theorem%3F+Grothendieck+and+the+Logic+of+Number+Theory&rft.volume=16&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E359-%3C%2Fspan%3E377&rft.date=2014-01-15&rft_id=info%3Adoi%2F10.2178%2Fbsl%2F1286284558&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13475845%23id-name%3DS2CID&rft.aulast=McLarty&rft.aufirst=Colin&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fjournals%2Fbulletin-of-symbolic-logic%2Farticle%2Fabs%2Fwhat-does-it-take-to-prove-fermats-last-theorem-grothendieck-and-the-logic-of-number-theory%2F80EDFF3616F8D58590EBA0DCB9FD2E3E&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFWallace2004">Wallace 2004</a>, p. 44</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAristotle" class="citation book cs1">Aristotle. <a rel="nofollow" class="external text" href="http://classics.mit.edu/Aristotle/physics.3.iii.html"><i>Physics</i></a>. Translated by Hardie, R. P.; Gaye, R. K. The Internet Classics Archive. Book 3, Chapters 5–8.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics&rft.pages=Book+3%2C+Chapters+5-8&rft.pub=The+Internet+Classics+Archive&rft.au=Aristotle&rft_id=http%3A%2F%2Fclassics.mit.edu%2FAristotle%2Fphysics.3.iii.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGoodman1981" class="citation book cs1">Goodman, Nicolas D. (1981). "Reflections on Bishop's philosophy of mathematics". In Richman, F. (ed.). <i>Constructive Mathematics</i>. Lecture Notes in Mathematics. Vol. 873. Springer. pp. <span class="nowrap">135–</span>145. <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%2FBFb0090732">10.1007/BFb0090732</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-10850-4" title="Special:BookSources/978-3-540-10850-4"><bdi>978-3-540-10850-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reflections+on+Bishop%27s+philosophy+of+mathematics&rft.btitle=Constructive+Mathematics&rft.series=Lecture+Notes+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E135-%3C%2Fspan%3E145&rft.pub=Springer&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2FBFb0090732&rft.isbn=978-3-540-10850-4&rft.aulast=Goodman&rft.aufirst=Nicolas+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></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">Maor, p. 3</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSarton1928" class="citation journal cs1">Sarton, George (March 1928). <a rel="nofollow" class="external text" href="https://www.journals.uchicago.edu/doi/10.1086/346308">"<i>The Thirteen Books of Euclid's Elements</i>. Thomas L. Heath, Heiberg"</a>. <i>Isis</i>. <b>10</b> (1): <span class="nowrap">60–</span>62. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F346308">10.1086/346308</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-1753">0021-1753</a> – via The University of Chicago Press Journals.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Isis&rft.atitle=The+Thirteen+Books+of+Euclid%27s+Elements.+Thomas+L.+Heath%2C+Heiberg&rft.volume=10&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E60-%3C%2Fspan%3E62&rft.date=1928-03&rft_id=info%3Adoi%2F10.1086%2F346308&rft.issn=0021-1753&rft.aulast=Sarton&rft.aufirst=George&rft_id=https%3A%2F%2Fwww.journals.uchicago.edu%2Fdoi%2F10.1086%2F346308&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHutten1962" class="citation book cs1">Hutten, Ernest Hirschlaff (1962). <a rel="nofollow" class="external text" href="https://archive.org/details/originsofscience0000hutt_n9u7"><i>The origins of science; an inquiry into the foundations of Western thought</i></a>. Internet Archive. London, Allen and Unwin. pp. <span class="nowrap">1–</span>241. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-04-946007-2" title="Special:BookSources/978-0-04-946007-2"><bdi>978-0-04-946007-2</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-01-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+origins+of+science%3B+an+inquiry+into+the+foundations+of+Western+thought&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E241&rft.pub=London%2C+Allen+and+Unwin&rft.date=1962&rft.isbn=978-0-04-946007-2&rft.aulast=Hutten&rft.aufirst=Ernest+Hirschlaff&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foriginsofscience0000hutt_n9u7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuclid2008" class="citation book cs1">Euclid (2008) [c. 300 BC]. <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf"><i>Euclid's Elements of Geometry</i></a> <span class="cs1-format">(PDF)</span>. Translated by Fitzpatrick, Richard. Lulu.com. p. 6 (Book I, Postulate 5). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-6151-7984-1" title="Special:BookSources/978-0-6151-7984-1"><bdi>978-0-6151-7984-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euclid%27s+Elements+of+Geometry&rft.pages=6+%28Book+I%2C+Postulate+5%29&rft.pub=Lulu.com&rft.date=2008&rft.isbn=978-0-6151-7984-1&rft.au=Euclid&rft_id=http%3A%2F%2Ffarside.ph.utexas.edu%2FBooks%2FEuclid%2FElements.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeathHeiberg1908" class="citation book cs1"><a href="/wiki/Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Heath, Sir Thomas Little</a>; Heiberg, Johan Ludvig (1908). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dkk6AQAAMAAJ&q=right+angles+infinite&pg=PR8"><i>The Thirteen Books of Euclid's Elements</i></a>. Vol. v. 1. The University Press. p. 212.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.pages=212&rft.pub=The+University+Press&rft.date=1908&rft.aulast=Heath&rft.aufirst=Sir+Thomas+Little&rft.au=Heiberg%2C+Johan+Ludvig&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Ddkk6AQAAMAAJ%26q%3Dright%2Bangles%2Binfinite%26pg%3DPR8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDrozdek2008" class="citation book cs1">Drozdek, Adam (2008). <i><span></span></i>In the Beginning Was the<i> Apeiron</i>: Infinity in Greek Philosophy<i><span></span></i>. Stuttgart, Germany: Franz Steiner Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-515-09258-6" title="Special:BookSources/978-3-515-09258-6"><bdi>978-3-515-09258-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=In+the+Beginning+Was+the+Apeiron%3A+Infinity+in+Greek+Philosophy&rft.place=Stuttgart%2C+Germany&rft.pub=Franz+Steiner+Verlag&rft.date=2008&rft.isbn=978-3-515-09258-6&rft.aulast=Drozdek&rft.aufirst=Adam&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-Zeno's_paradoxes-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zeno's_paradoxes_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/paradox-zeno/">"Zeno's Paradoxes"</a>. <i>Stanford University</i>. October 15, 2010<span class="reference-accessdate">. Retrieved <span class="nowrap">April 3,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stanford+University&rft.atitle=Zeno%27s+Paradoxes&rft.date=2010-10-15&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fparadox-zeno%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFRussell1996">Russell 1996</a>, p. 347</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCauchy1821" class="citation book cs1"><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy, Augustin-Louis</a> (1821). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UrT0KsbDmDwC&pg=PA1"><i>Cours d'Analyse de l'École Royale Polytechnique</i></a>. Libraires du Roi & de la Bibliothèque du Roi. p. 124<span class="reference-accessdate">. Retrieved <span class="nowrap">October 12,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cours+d%27Analyse+de+l%27%C3%89cole+Royale+Polytechnique&rft.pages=124&rft.pub=Libraires+du+Roi+%26+de+la+Biblioth%C3%A8que+du+Roi&rft.date=1821&rft.aulast=Cauchy&rft.aufirst=Augustin-Louis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUrT0KsbDmDwC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIan_Stewart2017" class="citation book cs1">Ian Stewart (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117"><i>Infinity: a Very Short Introduction</i></a>. Oxford University Press. p. 117. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-875523-4" title="Special:BookSources/978-0-19-875523-4"><bdi>978-0-19-875523-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170403200429/https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117">Archived</a> from the original on April 3, 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinity%3A+a+Very+Short+Introduction&rft.pages=117&rft.pub=Oxford+University+Press&rft.date=2017&rft.isbn=978-0-19-875523-4&rft.au=Ian+Stewart&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiewwDgAAQBAJ%26pg%3DPA117&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCajori2007" class="citation book cs1">Cajori, Florian (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OQZxHpG2y3UC&q=infinity"><i>A History of Mathematical Notations</i></a>. Vol. 1. Cosimo, Inc. p. 214. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781602066854" title="Special:BookSources/9781602066854"><bdi>9781602066854</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematical+Notations&rft.pages=214&rft.pub=Cosimo%2C+Inc.&rft.date=2007&rft.isbn=9781602066854&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOQZxHpG2y3UC%26q%3Dinfinity&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori1993">Cajori 1993</a>, Sec. 421, Vol. II, p. 44</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/ArithmeticaInfinitorum/page/n5/mode/2up">"Arithmetica Infinitorum"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Arithmetica+Infinitorum&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FArithmeticaInfinitorum%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori1993">Cajori 1993</a>, Sec. 435, Vol. II, p. 58</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrattan-Guinness2005" class="citation book cs1">Grattan-Guinness, Ivor (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UdGBy8iLpocC"><i>Landmark Writings in Western Mathematics 1640-1940</i></a>. Elsevier. p. 62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-045744-4" title="Special:BookSources/978-0-08-045744-4"><bdi>978-0-08-045744-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160603085825/https://books.google.com/books?id=UdGBy8iLpocC">Archived</a> from the original on 2016-06-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Landmark+Writings+in+Western+Mathematics+1640-1940&rft.pages=62&rft.pub=Elsevier&rft.date=2005&rft.isbn=978-0-08-045744-4&rft.aulast=Grattan-Guinness&rft.aufirst=Ivor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUdGBy8iLpocC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UdGBy8iLpocC&pg=PA62">Extract of p. 62</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeyl2012" class="citation cs2">Weyl, Hermann (2012), Peter Pesic (ed.), <i>Levels of Infinity / Selected Writings on Mathematics and Philosophy</i>, Dover, p. 17, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-48903-2" title="Special:BookSources/978-0-486-48903-2"><bdi>978-0-486-48903-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Levels+of+Infinity+%2F+Selected+Writings+on+Mathematics+and+Philosophy&rft.pages=17&rft.pub=Dover&rft.date=2012&rft.isbn=978-0-486-48903-2&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAG" class="citation web cs1">AG, Compart. <a rel="nofollow" class="external text" href="https://www.compart.com/en/unicode/U+221E">"Unicode Character "∞" (U+221E)"</a>. <i>Compart.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Compart.com&rft.atitle=Unicode+Character+%22%E2%88%9E%22+%28U%2B221E%29&rft.aulast=AG&rft.aufirst=Compart&rft_id=https%3A%2F%2Fwww.compart.com%2Fen%2Funicode%2FU%2B221E&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols">"List of LaTeX mathematical symbols - OeisWiki"</a>. <i>oeis.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=oeis.org&rft.atitle=List+of+LaTeX+mathematical+symbols+-+OeisWiki&rft_id=https%3A%2F%2Foeis.org%2Fwiki%2FList_of_LaTeX_mathematical_symbols&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFScott1981" class="citation cs2">Scott, Joseph Frederick (1981), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24"><i>The mathematical work of John Wallis, D.D., F.R.S., (1616–1703)</i></a> (2 ed.), <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, p. 24, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8284-0314-6" title="Special:BookSources/978-0-8284-0314-6"><bdi>978-0-8284-0314-6</bdi></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160509151853/https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24">archived</a> from the original on 2016-05-09</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+mathematical+work+of+John+Wallis%2C+D.D.%2C+F.R.S.%2C+%281616%E2%80%931703%29&rft.pages=24&rft.edition=2&rft.pub=American+Mathematical+Society&rft.date=1981&rft.isbn=978-0-8284-0314-6&rft.aulast=Scott&rft.aufirst=Joseph+Frederick&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXX9PKytw8g8C%26pg%3DPA24&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMartin-Löf1990" class="citation cs2"><a href="/wiki/Per_Martin-L%C3%B6f" title="Per Martin-Löf">Martin-Löf, Per</a> (1990), "Mathematics of infinity", <i>COLOG-88 (Tallinn, 1988)</i>, Lecture Notes in Computer Science, vol. 417, Berlin: Springer, pp. <span class="nowrap">146–</span>197, <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%2F3-540-52335-9_54">10.1007/3-540-52335-9_54</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-52335-2" title="Special:BookSources/978-3-540-52335-2"><bdi>978-3-540-52335-2</bdi></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=1064143">1064143</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+of+infinity&rft.btitle=COLOG-88+%28Tallinn%2C+1988%29&rft.place=Berlin&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E146-%3C%2Fspan%3E197&rft.pub=Springer&rft.date=1990&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1064143%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F3-540-52335-9_54&rft.isbn=978-3-540-52335-2&rft.aulast=Martin-L%C3%B6f&rft.aufirst=Per&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFO'Flaherty1986" class="citation cs2">O'Flaherty, Wendy Doniger (1986), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243"><i>Dreams, Illusion, and Other Realities</i></a>, University of Chicago Press, p. 243, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-61855-5" title="Special:BookSources/978-0-226-61855-5"><bdi>978-0-226-61855-5</bdi></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160629143323/https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243">archived</a> from the original on 2016-06-29</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dreams%2C+Illusion%2C+and+Other+Realities&rft.pages=243&rft.pub=University+of+Chicago+Press&rft.date=1986&rft.isbn=978-0-226-61855-5&rft.aulast=O%27Flaherty&rft.aufirst=Wendy+Doniger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvhNNrX3bmo4C%26pg%3DPA243&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFToker1989" class="citation cs2">Toker, Leona (1989), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159"><i>Nabokov: The Mystery of Literary Structures</i></a>, Cornell University Press, p. 159, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8014-2211-9" title="Special:BookSources/978-0-8014-2211-9"><bdi>978-0-8014-2211-9</bdi></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160509095701/https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159">archived</a> from the original on 2016-05-09</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nabokov%3A+The+Mystery+of+Literary+Structures&rft.pages=159&rft.pub=Cornell+University+Press&rft.date=1989&rft.isbn=978-0-8014-2211-9&rft.aulast=Toker&rft.aufirst=Leona&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJud1q_NrqpcC%26pg%3DPA159&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBell" class="citation encyclopaedia cs1"><a href="/wiki/John_Lane_Bell" title="John Lane Bell">Bell, John Lane</a>. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/continuity/">"Continuity and Infinitesimals"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Continuity+and+Infinitesimals&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.aulast=Bell&rft.aufirst=John+Lane&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fcontinuity%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><a href="#CITEREFTaylor1955">Taylor 1955</a>, p. 63</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">These uses of infinity for integrals and series can be found in any standard calculus text, such as, <a href="#CITEREFSwokowski1983">Swokowski 1983</a>, pp. 468–510</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mathonline.wikidot.com/properly-divergent-sequences">"Properly Divergent Sequences - Mathonline"</a>. <i>mathonline.wikidot.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathonline.wikidot.com&rft.atitle=Properly+Divergent+Sequences+-+Mathonline&rft_id=http%3A%2F%2Fmathonline.wikidot.com%2Fproperly-divergent-sequences&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAliprantisBurkinshaw1998" class="citation cs2">Aliprantis, Charalambos D.; Burkinshaw, Owen (1998), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=m40ivUwAonUC&pg=PA29"><i>Principles of Real Analysis</i></a> (3rd ed.), San Diego, CA: Academic Press, Inc., p. 29, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-050257-8" title="Special:BookSources/978-0-12-050257-8"><bdi>978-0-12-050257-8</bdi></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=1669668">1669668</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150515120230/https://books.google.com/books?id=m40ivUwAonUC&pg=PA29">archived</a> from the original on 2015-05-15</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Real+Analysis&rft.place=San+Diego%2C+CA&rft.pages=29&rft.edition=3rd&rft.pub=Academic+Press%2C+Inc.&rft.date=1998&rft.isbn=978-0-12-050257-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1669668%23id-name%3DMR&rft.aulast=Aliprantis&rft.aufirst=Charalambos+D.&rft.au=Burkinshaw%2C+Owen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dm40ivUwAonUC%26pg%3DPA29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><a href="#CITEREFGemignani1990">Gemignani 1990</a>, p. 177</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBeutelspacherRosenbaum1998" class="citation cs2">Beutelspacher, Albrecht; Rosenbaum, Ute (1998), <i>Projective Geometry / from foundations to applications</i>, Cambridge University Press, p. 27, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-48364-3" title="Special:BookSources/978-0-521-48364-3"><bdi>978-0-521-48364-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+Geometry+%2F+from+foundations+to+applications&rft.pages=27&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=978-0-521-48364-3&rft.aulast=Beutelspacher&rft.aufirst=Albrecht&rft.au=Rosenbaum%2C+Ute&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRaoStetkær1991" class="citation book cs1">Rao, Murali; Stetkær, Henrik (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wdTntZ_N0tYC&pg=PA113"><i>Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory</i></a>. World Scientific. p. 113. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789810203757" title="Special:BookSources/9789810203757"><bdi>9789810203757</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis%3A+An+Invitation+%3A+a+Concise+Introduction+to+Complex+Function+Theory&rft.pages=113&rft.pub=World+Scientific&rft.date=1991&rft.isbn=9789810203757&rft.aulast=Rao&rft.aufirst=Murali&rft.au=Stetk%C3%A6r%2C+Henrik&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwdTntZ_N0tYC%26pg%3DPA113&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEric_Schechter2005" class="citation book cs1">Eric Schechter (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=70W4Q-kzdicC"><i>Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions</i></a> (illustrated ed.). Princeton University Press. p. 118. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12279-3" title="Special:BookSources/978-0-691-12279-3"><bdi>978-0-691-12279-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+and+Nonclassical+Logics%3A+An+Introduction+to+the+Mathematics+of+Propositions&rft.pages=118&rft.edition=illustrated&rft.pub=Princeton+University+Press&rft.date=2005&rft.isbn=978-0-691-12279-3&rft.au=Eric+Schechter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D70W4Q-kzdicC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=70W4Q-kzdicC&pg=PA118">Extract of page 118</a></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFA.W._Moore2012" class="citation book cs1">A.W. Moore (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z-UJhZmQnhAC"><i>The Infinite</i></a> (2nd, revised ed.). Routledge. p. xiv. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-134-91213-1" title="Special:BookSources/978-1-134-91213-1"><bdi>978-1-134-91213-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Infinite&rft.pages=xiv&rft.edition=2nd%2C+revised&rft.pub=Routledge&rft.date=2012&rft.isbn=978-1-134-91213-1&rft.au=A.W.+Moore&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dz-UJhZmQnhAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z-UJhZmQnhAC&pg=PR14">Extract of page xiv</a></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRudolf_V_Rucker2019" class="citation book cs1">Rudolf V Rucker (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jv2EDwAAQBAJ"><i>Infinity and the Mind: The Science and Philosophy of the Infinite</i></a> (illustrated ed.). Princeton University Press. p. 85. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-19125-6" title="Special:BookSources/978-0-691-19125-6"><bdi>978-0-691-19125-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinity+and+the+Mind%3A+The+Science+and+Philosophy+of+the+Infinite&rft.pages=85&rft.edition=illustrated&rft.pub=Princeton+University+Press&rft.date=2019&rft.isbn=978-0-691-19125-6&rft.au=Rudolf+V+Rucker&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Djv2EDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jv2EDwAAQBAJ&pg=PA85">Extract of page 85</a></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDauben1993" class="citation journal cs1">Dauben, Joseph (1993). <a rel="nofollow" class="external text" href="http://acmsonline.org/home2/wp-content/uploads/2016/05/Dauben-Cantor.pdf">"Georg Cantor and the Battle for Transfinite Set Theory"</a> <span class="cs1-format">(PDF)</span>. <i>9th ACMS Conference Proceedings</i>: 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=9th+ACMS+Conference+Proceedings&rft.atitle=Georg+Cantor+and+the+Battle+for+Transfinite+Set+Theory&rft.pages=4&rft.date=1993&rft.aulast=Dauben&rft.aufirst=Joseph&rft_id=http%3A%2F%2Facmsonline.org%2Fhome2%2Fwp-content%2Fuploads%2F2016%2F05%2FDauben-Cantor.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><a href="#CITEREFCohen1963">Cohen 1963</a>, p. 1143</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFelix_Hausdorff2021" class="citation book cs1">Felix Hausdorff (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TFA_EAAAQBAJ"><i>Set Theory</i></a>. American Mathematical Soc. p. 44. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-6494-3" title="Special:BookSources/978-1-4704-6494-3"><bdi>978-1-4704-6494-3</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.pages=44&rft.pub=American+Mathematical+Soc.&rft.date=2021&rft.isbn=978-1-4704-6494-3&rft.au=Felix+Hausdorff&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTFA_EAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TFA_EAAAQBAJ&pg=PA44">Extract of page 44</a></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><a href="#CITEREFSagan1994">Sagan 1994</a>, pp. 10–12</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMichael_FrameBenoit_Mandelbrot2002" class="citation book cs1">Michael Frame; Benoit Mandelbrot (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Wz7iCaiB2C0C"><i>Fractals, Graphics, and Mathematics Education</i></a> (illustrated ed.). Cambridge University Press. p. 36. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-169-2" title="Special:BookSources/978-0-88385-169-2"><bdi>978-0-88385-169-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractals%2C+Graphics%2C+and+Mathematics+Education&rft.pages=36&rft.edition=illustrated&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-88385-169-2&rft.au=Michael+Frame&rft.au=Benoit+Mandelbrot&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWz7iCaiB2C0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Wz7iCaiB2C0C&pg=PA36">Extract of page 36</a></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><a href="#CITEREFKline1972">Kline 1972</a>, pp. 1197–1198</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.doriclenses.com/administrer/upload/pdf/NOT_AXI_ENG_070212_doricl97_doricle_kvgwQP.pdf">Doric Lenses</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130124011604/http://www.doriclenses.com/administrer/upload/pdf/NOT_AXI_ENG_070212_doricl97_doricle_kvgwQP.pdf">Archived</a> 2013-01-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">John Gribbin (2009), <i>In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality</i>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-61352-8" title="Special:BookSources/978-0-470-61352-8">978-0-470-61352-8</a>. p. 88</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrake2013" class="citation book cs1">Brake, Mark (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sWGqzfL0snEC&pg=PA63"><i>Alien Life Imagined: Communicating the Science and Culture of Astrobiology</i></a> (illustrated ed.). Cambridge University Press. p. 63. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-49129-7" title="Special:BookSources/978-0-521-49129-7"><bdi>978-0-521-49129-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alien+Life+Imagined%3A+Communicating+the+Science+and+Culture+of+Astrobiology&rft.pages=63&rft.edition=illustrated&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-0-521-49129-7&rft.aulast=Brake&rft.aufirst=Mark&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsWGqzfL0snEC%26pg%3DPA63&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoupelisKuhn2007" class="citation book cs1">Koupelis, Theo; Kuhn, Karl F. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6rTttN4ZdyoC"><i>In Quest of the Universe</i></a> (illustrated ed.). Jones & Bartlett Learning. p. 553. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-4387-1" title="Special:BookSources/978-0-7637-4387-1"><bdi>978-0-7637-4387-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=In+Quest+of+the+Universe&rft.pages=553&rft.edition=illustrated&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2007&rft.isbn=978-0-7637-4387-1&rft.aulast=Koupelis&rft.aufirst=Theo&rft.au=Kuhn%2C+Karl+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6rTttN4ZdyoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6rTttN4ZdyoC&pg=PA553">Extract of p. 553</a></span> </li> <li id="cite_note-NASA_Shape-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-NASA_Shape_52-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://map.gsfc.nasa.gov/universe/uni_shape.html">"Will the Universe expand forever?"</a>. NASA. 24 January 2014. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120601032707/http://map.gsfc.nasa.gov/universe/uni_shape.html">Archived</a> from the original on 1 June 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">16 March</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Will+the+Universe+expand+forever%3F&rft.pub=NASA&rft.date=2014-01-24&rft_id=http%3A%2F%2Fmap.gsfc.nasa.gov%2Funiverse%2Funi_shape.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-Fermi_Flat-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-Fermi_Flat_53-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725">"Our universe is Flat"</a>. FermiLab/SLAC. 7 April 2015. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150410200411/http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725">Archived</a> from the original on 10 April 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Our+universe+is+Flat&rft.pub=FermiLab%2FSLAC&rft.date=2015-04-07&rft_id=http%3A%2F%2Fwww.symmetrymagazine.org%2Farticle%2Fapril-2015%2Four-flat-universe%3Femail_issue%3D725&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMarcus_Y._Yoo2011" class="citation journal cs1 cs1-prop-long-vol">Marcus Y. Yoo (2011). "Unexpected connections". <i>Engineering & Science</i>. LXXIV1: 30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Engineering+%26+Science&rft.atitle=Unexpected+connections&rft.volume=LXXIV1&rft.pages=30&rft.date=2011&rft.au=Marcus+Y.+Yoo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeeks2001" class="citation book cs1">Weeks, Jeffrey (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/shapeofspace0000week"><i>The Shape of Space</i></a></span>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-0709-5" title="Special:BookSources/978-0-8247-0709-5"><bdi>978-0-8247-0709-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Shape+of+Space&rft.pub=CRC+Press&rft.date=2001&rft.isbn=978-0-8247-0709-5&rft.aulast=Weeks&rft.aufirst=Jeffrey&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fshapeofspace0000week&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text">Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.</span> </li> <li id="cite_note-Nautilus2014-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nautilus2014_57-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="Chapter_4" class="citation news cs1">McKee, Maggie (25 September 2014). <a rel="nofollow" class="external text" href="http://nautil.us/issue/17/big-bangs/ingenious-paul-j-steinhardt">"Ingenious: Paul J. Steinhardt – The Princeton physicist on what's wrong with inflation theory and his view of the Big Bang"</a>. <i>Nautilus</i>. No. 17. NautilusThink Inc<span class="reference-accessdate">. Retrieved <span class="nowrap">31 March</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nautilus&rft.atitle=Ingenious%3A+Paul+J.+Steinhardt+%E2%80%93+The+Princeton+physicist+on+what%27s+wrong+with+inflation+theory+and+his+view+of+the+Big+Bang&rft.issue=17&rft.date=2014-09-25&rft.aulast=McKee&rft.aufirst=Maggie&rft_id=http%3A%2F%2Fnautil.us%2Fissue%2F17%2Fbig-bangs%2Fingenious-paul-j-steinhardt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><i>Cambridge Dictionary of Philosophy</i>, Second Edition, p. 429</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.gnu.org/software/libc/manual/html_node/Infinity-and-NaN.html">"Infinity and NaN (The GNU C Library)"</a>. <i>www.gnu.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-03-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.gnu.org&rft.atitle=Infinity+and+NaN+%28The+GNU+C+Library%29&rft_id=https%3A%2F%2Fwww.gnu.org%2Fsoftware%2Flibc%2Fmanual%2Fhtml_node%2FInfinity-and-NaN.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGosling2012" class="citation book cs1">Gosling, James; et al. (27 July 2012). <a rel="nofollow" class="external text" href="http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3">"4.2.3."</a>. <i>The Java Language Specification</i> (Java SE 7 ed.). California: Oracle America, Inc. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120609071157/http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3">Archived</a> from the original on 9 June 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">6 September</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=4.2.3.&rft.btitle=The+Java+Language+Specification&rft.place=California&rft.edition=Java+SE+7&rft.pub=Oracle+America%2C+Inc.&rft.date=2012-07-27&rft.aulast=Gosling&rft.aufirst=James&rft_id=http%3A%2F%2Fdocs.oracle.com%2Fjavase%2Fspecs%2Fjls%2Fse7%2Fhtml%2Fjls-4.html%23jls-4.2.3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStokes2012" class="citation book cs1">Stokes, Roger (July 2012). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120325064205/http://www.rogerstokes.free-online.co.uk/19.htm#10">"19.2.1"</a>. <i>Learning J</i>. Archived from <a rel="nofollow" class="external text" href="http://www.rogerstokes.free-online.co.uk/19.htm#10">the original</a> on 25 March 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">6 September</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=19.2.1&rft.btitle=Learning+J&rft.date=2012-07&rft.aulast=Stokes&rft.aufirst=Roger&rft_id=http%3A%2F%2Fwww.rogerstokes.free-online.co.uk%2F19.htm%2310&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMorris_Kline1985" class="citation book cs1">Morris Kline (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gXSMukf1aFIC"><i>Mathematics for the Nonmathematician</i></a> (illustrated, unabridged, reprinted ed.). Courier Corporation. p. 229. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-24823-3" title="Special:BookSources/978-0-486-24823-3"><bdi>978-0-486-24823-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+the+Nonmathematician&rft.pages=229&rft.edition=illustrated%2C+unabridged%2C+reprinted&rft.pub=Courier+Corporation&rft.date=1985&rft.isbn=978-0-486-24823-3&rft.au=Morris+Kline&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgXSMukf1aFIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gXSMukf1aFIC&pg=PA229">Extract of page 229, Section 10-7</a></span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchattschneider2010" class="citation journal cs1">Schattschneider, Doris (2010). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/201006/rtx100600706p.pdf">"The Mathematical Side of M. C. Escher"</a> <span class="cs1-format">(PDF)</span>. <i>Notices of the AMS</i>. <b>57</b> (6): <span class="nowrap">706–</span>718.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=The+Mathematical+Side+of+M.+C.+Escher&rft.volume=57&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E706-%3C%2Fspan%3E718&rft.date=2010&rft.aulast=Schattschneider&rft.aufirst=Doris&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F201006%2Frtx100600706p.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.chessvariants.com/boardrules.dir/infinite.html">Infinite chess at the Chess Variant Pages</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170402082426/http://www.chessvariants.com/boardrules.dir/infinite.html">Archived</a> 2017-04-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> An infinite chess scheme.</span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=PN-I6u-AxMg">"Infinite Chess, PBS Infinite Series"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170407211614/https://www.youtube.com/watch?v=PN-I6u-AxMg">Archived</a> 2017-04-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> PBS Infinite Series, with academic sources by J. Hamkins (infinite chess: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEvansJoel_David_Hamkins2013" class="citation arxiv cs1">Evans, C.D.A; Joel David Hamkins (2013). "Transfinite game values in infinite chess". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1302.4377">1302.4377</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Transfinite+game+values+in+infinite+chess&rft.date=2013&rft_id=info%3Aarxiv%2F1302.4377&rft.aulast=Evans&rft.aufirst=C.D.A&rft.au=Joel+David+Hamkins&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEvansJoel_David_HamkinsNorman_Lewis_Perlmutter2015" class="citation arxiv cs1">Evans, C.D.A; Joel David Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $ω^4$". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1510.08155">1510.08155</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+position+in+infinite+chess+with+game+value+%24%CF%89%5E4%24&rft.date=2015&rft_id=info%3Aarxiv%2F1510.08155&rft.aulast=Evans&rft.aufirst=C.D.A&rft.au=Joel+David+Hamkins&rft.au=Norman+Lewis+Perlmutter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span>).</span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFElglalyQuek" class="citation web cs1">Elglaly, Yasmine Nader; Quek, Francis. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200226004335/http://www.se.rit.edu/~yasmine/assets/papers/Embodied%20math.pdf">"Review of "Where Mathematics comes from: How the Embodied Mind Brings Mathematics Into Being" By George Lakoff and Rafael E. Nunez"</a> <span class="cs1-format">(PDF)</span>. <i>CHI 2009</i>. Archived from <a rel="nofollow" class="external text" href="http://www.se.rit.edu/~yasmine/assets/papers/Embodied%20math.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2020-02-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-03-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CHI+2009&rft.atitle=Review+of+%22Where+Mathematics+comes+from%3A+How+the+Embodied+Mind+Brings+Mathematics+Into+Being%22+By+George+Lakoff+and+Rafael+E.+Nunez&rft.aulast=Elglaly&rft.aufirst=Yasmine+Nader&rft.au=Quek%2C+Francis&rft_id=http%3A%2F%2Fwww.se.rit.edu%2F~yasmine%2Fassets%2Fpapers%2FEmbodied%2520math.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=25" title="Edit section: Bibliography"><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" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCajori1993" class="citation cs2">Cajori, Florian (1993) [1928 & 1929], <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00cajo_0"><i>A History of Mathematical Notations (Two Volumes Bound as One)</i></a>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67766-8" title="Special:BookSources/978-0-486-67766-8"><bdi>978-0-486-67766-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematical+Notations+%28Two+Volumes+Bound+as+One%29&rft.pub=Dover&rft.date=1993&rft.isbn=978-0-486-67766-8&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00cajo_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGemignani1990" class="citation cs2">Gemignani, Michael C. (1990), <i>Elementary Topology</i> (2nd ed.), Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66522-1" title="Special:BookSources/978-0-486-66522-1"><bdi>978-0-486-66522-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Topology&rft.edition=2nd&rft.pub=Dover&rft.date=1990&rft.isbn=978-0-486-66522-1&rft.aulast=Gemignani&rft.aufirst=Michael+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKeisler1986" class="citation cs2"><a href="/wiki/Howard_Jerome_Keisler" title="Howard Jerome Keisler">Keisler, H. Jerome</a> (1986), <a rel="nofollow" class="external text" href="http://www.math.wisc.edu/~keisler/calc.html"><i>Elementary Calculus: An Approach Using Infinitesimals</i></a> (2nd ed.)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Calculus%3A+An+Approach+Using+Infinitesimals&rft.edition=2nd&rft.date=1986&rft.aulast=Keisler&rft.aufirst=H.+Jerome&rft_id=http%3A%2F%2Fwww.math.wisc.edu%2F~keisler%2Fcalc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaddox2002" class="citation cs2">Maddox, Randall B. (2002), <i>Mathematical Thinking and Writing: A Transition to Abstract Mathematics</i>, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-464976-7" title="Special:BookSources/978-0-12-464976-7"><bdi>978-0-12-464976-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thinking+and+Writing%3A+A+Transition+to+Abstract+Mathematics&rft.pub=Academic+Press&rft.date=2002&rft.isbn=978-0-12-464976-7&rft.aulast=Maddox&rft.aufirst=Randall+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKline1972" class="citation cs2"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a> (1972), <i>Mathematical Thought from Ancient to Modern Times</i>, New York: Oxford University Press, pp. <span class="nowrap">1197–</span>1198, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-506135-2" title="Special:BookSources/978-0-19-506135-2"><bdi>978-0-19-506135-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thought+from+Ancient+to+Modern+Times&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1197-%3C%2Fspan%3E1198&rft.pub=Oxford+University+Press&rft.date=1972&rft.isbn=978-0-19-506135-2&rft.aulast=Kline&rft.aufirst=Morris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRussell1996" class="citation cs2"><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell, Bertrand</a> (1996) [1903], <i>The Principles of Mathematics</i>, New York: Norton, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-31404-5" title="Special:BookSources/978-0-393-31404-5"><bdi>978-0-393-31404-5</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/247299160">247299160</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Principles+of+Mathematics&rft.place=New+York&rft.pub=Norton&rft.date=1996&rft_id=info%3Aoclcnum%2F247299160&rft.isbn=978-0-393-31404-5&rft.aulast=Russell&rft.aufirst=Bertrand&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSagan1994" class="citation cs2">Sagan, Hans (1994), <i>Space-Filling Curves</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-0871-6" title="Special:BookSources/978-1-4612-0871-6"><bdi>978-1-4612-0871-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space-Filling+Curves&rft.pub=Springer&rft.date=1994&rft.isbn=978-1-4612-0871-6&rft.aulast=Sagan&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSwokowski1983" class="citation cs2">Swokowski, Earl W. (1983), <a rel="nofollow" class="external text" href="https://archive.org/details/calculuswithanal00swok"><i>Calculus with Analytic Geometry</i></a> (Alternate ed.), Prindle, Weber & Schmidt, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-87150-341-1" title="Special:BookSources/978-0-87150-341-1"><bdi>978-0-87150-341-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+with+Analytic+Geometry&rft.edition=Alternate&rft.pub=Prindle%2C+Weber+%26+Schmidt&rft.date=1983&rft.isbn=978-0-87150-341-1&rft.aulast=Swokowski&rft.aufirst=Earl+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculuswithanal00swok&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTaylor1955" class="citation cs2">Taylor, Angus E. (1955), <i>Advanced Calculus</i>, Blaisdell Publishing Company</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Calculus&rft.pub=Blaisdell+Publishing+Company&rft.date=1955&rft.aulast=Taylor&rft.aufirst=Angus+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWallace2004" class="citation cs2"><a href="/wiki/David_Foster_Wallace" title="David Foster Wallace">Wallace, David Foster</a> (2004), <i>Everything and More: A Compact History of Infinity</i>, Norton, W.W. & Company, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-32629-1" title="Special:BookSources/978-0-393-32629-1"><bdi>978-0-393-32629-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Everything+and+More%3A+A+Compact+History+of+Infinity&rft.pub=Norton%2C+W.W.+%26+Company%2C+Inc.&rft.date=2004&rft.isbn=978-0-393-32629-1&rft.aulast=Wallace&rft.aufirst=David+Foster&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinity&action=edit&section=26" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316" /><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAczel2001" class="citation book cs1">Aczel, Amir D. (2001). <i>The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity</i>. New York: Pocket Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7434-2299-4" title="Special:BookSources/978-0-7434-2299-4"><bdi>978-0-7434-2299-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mystery+of+the+Aleph%3A+Mathematics%2C+the+Kabbalah%2C+and+the+Search+for+Infinity&rft.place=New+York&rft.pub=Pocket+Books&rft.date=2001&rft.isbn=978-0-7434-2299-4&rft.aulast=Aczel&rft.aufirst=Amir+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><a href="/wiki/D.P._Agrawal" class="mw-redirect" title="D.P. Agrawal">D.P. Agrawal</a> (2000). <i><a rel="nofollow" class="external text" href="http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina.htm">Ancient Jaina Mathematics: an Introduction</a></i>, <a rel="nofollow" class="external text" href="http://infinityfoundation.com">Infinity Foundation</a>.</li> <li>Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCohen1963" class="citation cs2">Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", <i><a href="/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America" title="Proceedings of the National Academy of Sciences of the United States of America">Proceedings of the National Academy of Sciences of the United States of America</a></i>, <b>50</b> (6): <span class="nowrap">1143–</span>1148, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963PNAS...50.1143C">1963PNAS...50.1143C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.50.6.1143">10.1073/pnas.50.6.1143</a></span>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287">221287</a></span>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578557">16578557</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=The+Independence+of+the+Continuum+Hypothesis&rft.volume=50&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1143-%3C%2Fspan%3E1148&rft.date=1963&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC221287%23id-name%3DPMC&rft_id=info%3Apmid%2F16578557&rft_id=info%3Adoi%2F10.1073%2Fpnas.50.6.1143&rft_id=info%3Abibcode%2F1963PNAS...50.1143C&rft.aulast=Cohen&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJain1982" class="citation book cs1">Jain, L.C. (1982). <i>Exact Sciences from Jaina Sources</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exact+Sciences+from+Jaina+Sources&rft.date=1982&rft.aulast=Jain&rft.aufirst=L.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li>Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", <i>Indian Journal of History of Science</i>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoseph2000" class="citation book cs1">Joseph, George G. (2000). <i>The Crest of the Peacock: Non-European Roots of Mathematics</i> (2nd ed.). <a href="/wiki/Penguin_Books" title="Penguin Books">Penguin Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-14-027778-4" title="Special:BookSources/978-0-14-027778-4"><bdi>978-0-14-027778-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Crest+of+the+Peacock%3A+Non-European+Roots+of+Mathematics&rft.edition=2nd&rft.pub=Penguin+Books&rft.date=2000&rft.isbn=978-0-14-027778-4&rft.aulast=Joseph&rft.aufirst=George+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li>H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at <a rel="nofollow" class="external free" href="http://www.math.wisc.edu/~keisler/calc.html">http://www.math.wisc.edu/~keisler/calc.html</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEli_Maor1991" class="citation book cs1"><a href="/wiki/Eli_Maor" title="Eli Maor">Eli Maor</a> (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lXjF7JnHQoIC&q=To+Infinity+and+beyond"><i>To Infinity and Beyond</i></a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02511-7" title="Special:BookSources/978-0-691-02511-7"><bdi>978-0-691-02511-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=To+Infinity+and+Beyond&rft.pub=Princeton+University+Press&rft.date=1991&rft.isbn=978-0-691-02511-7&rft.au=Eli+Maor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlXjF7JnHQoIC%26q%3DTo%2BInfinity%2Band%2Bbeyond&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li>O'Connor, John J. and Edmund F. Robertson (1998). <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html">'Georg Ferdinand Ludwig Philipp Cantor'</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060916095918/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html">Archived</a> 2006-09-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a></i>.</li> <li>O'Connor, John J. and Edmund F. Robertson (2000). <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html">'Jaina mathematics'</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081220145242/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html">Archived</a> 2008-12-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <i>MacTutor History of Mathematics archive</i>.</li> <li>Pearce, Ian. (2002). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html">'Jainism'</a>, <i>MacTutor History of Mathematics archive</i>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRucker1995" class="citation book cs1"><a href="/wiki/Rudy_Rucker" title="Rudy Rucker">Rucker, Rudy</a> (1995). <i>Infinity and the Mind: The Science and Philosophy of the Infinite</i>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-00172-2" title="Special:BookSources/978-0-691-00172-2"><bdi>978-0-691-00172-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinity+and+the+Mind%3A+The+Science+and+Philosophy+of+the+Infinite&rft.pub=Princeton+University+Press&rft.date=1995&rft.isbn=978-0-691-00172-2&rft.aulast=Rucker&rft.aufirst=Rudy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSingh1988" class="citation journal cs1">Singh, Navjyoti (1988). "Jaina Theory of Actual Infinity and Transfinite Numbers". <i>Journal of the Asiatic Society</i>. <b>30</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Asiatic+Society&rft.atitle=Jaina+Theory+of+Actual+Infinity+and+Transfinite+Numbers&rft.volume=30&rft.date=1988&rft.aulast=Singh&rft.aufirst=Navjyoti&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></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=Infinity&action=edit&section=27" 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 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src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/120px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Infinity_is_not_a_number" class="extiw" title="wikibooks:Infinity is not a number">Infinity is not a number</a></b></i></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 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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"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Infinity" class="extiw" title="q:Special:Search/Infinity">Infinity</a></b></i>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/infinite">"The Infinite"</a>. <i><a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Infinite&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Finfinite&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/p0054927">Infinity</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)"><i>In Our Time</i></a> at the <a href="/wiki/BBC" title="BBC">BBC</a></li> <li><i><a rel="nofollow" class="external text" href="http://www.earlham.edu/~peters/writing/infapp.htm">A Crash Course in the Mathematics of Infinite Sets</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100227033849/http://www.earlham.edu/~peters/writing/infapp.htm">Archived</a> 2010-02-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to <i>Infinite Reflections</i>, below. A concise introduction to Cantor's mathematics of infinite sets.</li> <li><i><a rel="nofollow" class="external text" href="http://www.earlham.edu/~peters/writing/infinity.htm">Infinite Reflections</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091105182928/http://www.earlham.edu/~peters/writing/infinity.htm">Archived</a> 2009-11-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrime" class="citation web cs1">Grime, James. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171022173525/http://www.numberphile.com/videos/countable_infinity.html">"Infinity is bigger than you think"</a>. <i>Numberphile</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="http://www.numberphile.com/videos/countable_infinity.html">the original</a> on 2017-10-22<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=Infinity+is+bigger+than+you+think&rft.aulast=Grime&rft.aufirst=James&rft_id=http%3A%2F%2Fwww.numberphile.com%2Fvideos%2Fcountable_infinity.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInfinity" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040910082530/http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html">Hotel Infinity</a></li> <li>John J. O'Connor and Edmund F. Robertson (1998). <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html">'Georg Ferdinand Ludwig Philipp Cantor'</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060916095918/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html">Archived</a> 2006-09-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a></i>.</li> <li>John J. O'Connor and Edmund F. Robertson (2000). <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html">'Jaina mathematics'</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081220145242/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html">Archived</a> 2008-12-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <i>MacTutor History of Mathematics archive</i>.</li> <li>Ian Pearce (2002). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html">'Jainism'</a>, <i>MacTutor History of Mathematics archive</i>.</li> <li><a rel="nofollow" class="external text" href="https://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/mysteryaleph.htm">The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity</a></li> <li><a rel="nofollow" class="external text" href="http://dictionary.of-the-infinite.com">Dictionary of the Infinite</a> (compilation of articles about infinity in physics, mathematics, and philosophy)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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selflink">Infinity</a> (<span class="texhtml"><a href="/wiki/Infinity_symbol" title="Infinity symbol">∞</a></span>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">History</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/Ananta_(infinite)" title="Ananta (infinite)">Ananta (infinite)</a></li> <li><a href="/wiki/Apeiron" title="Apeiron">Apeiron</a></li> <li><a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">Controversy over Cantor's theory</a></li> <li><a href="/wiki/Galileo%27s_paradox" title="Galileo's paradox">Galileo's paradox</a></li> <li><a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's paradox of the Grand Hotel</a></li> <li><a href="/wiki/Infinity_(philosophy)" title="Infinity (philosophy)">Infinity (philosophy)</a></li> <li><a href="/wiki/Paradoxes_of_infinity" class="mw-redirect" title="Paradoxes of infinity">Paradoxes of infinity</a></li> <li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes of set theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Branches of mathematics</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/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Internal_set_theory" title="Internal set theory">Internal set theory</a></li> <li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Synthetic_differential_geometry" title="Synthetic differential geometry">Synthetic differential geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formalizations of infinity</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/0.999..." title="0.999...">0.999...</a></li> <li><a href="/wiki/Absolute_infinite" title="Absolute infinite">Absolute infinite</a></li> <li><a href="/wiki/Actual_infinity" title="Actual infinity">Actual infinity</a></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Beth_number" title="Beth number">Beth number</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">Cardinality of the continuum</a></li> <li><a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite set</a></li> <li><a href="/wiki/Directed_infinity" title="Directed infinity">Directed infinity</a></li> <li><a href="/wiki/Division_by_zero" title="Division by zero">Division by zero</a> (Complex infinity)</li> <li><a href="/wiki/Epsilon_number" title="Epsilon number">Epsilon number</a></li> <li><a href="/wiki/Gimel_function" title="Gimel function">Gimel function</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite set</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Point_at_infinity" title="Point at infinity">Point at infinity</a></li> <li><a href="/wiki/Regular_cardinal" title="Regular cardinal">Regular cardinal</a></li> <li><a href="/wiki/Sphere_at_infinity" class="mw-redirect" title="Sphere at infinity">Sphere at infinity</a> (Kleinian group)</li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Transfinite_number" title="Transfinite number">Transfinite numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Geometries</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/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">Differential geometry of surfaces</a></li> <li><a href="/wiki/M%C3%B6bius_plane" title="Möbius plane">Möbius plane</a></li> <li><a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</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/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Large_numbers201" style="padding:3px"><table class="nowraplinks hlist 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:Large_numbers" title="Template:Large numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Large_numbers" title="Template talk:Large numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Large_numbers" title="Special:EditPage/Template:Large numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Large_numbers201" style="font-size:114%;margin:0 4em"><a href="/wiki/Large_numbers" title="Large numbers">Large numbers</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples <br />in<br />numerical <br />order</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/100" title="100">Hundred</a></li> <li><a href="/wiki/1000_(number)" title="1000 (number)">Thousand</a></li> <li><a href="/wiki/10,000" title="10,000">Ten thousand</a></li> <li><a href="/wiki/100,000" title="100,000">Hundred thousand</a></li> <li><a href="/wiki/1,000,000" title="1,000,000">Million</a></li> <li><a href="/wiki/1,000,000,000" title="1,000,000,000">Billion</a></li> <li><a href="/wiki/Trillion" title="Trillion">Trillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1015" title="Orders of magnitude (numbers)">Quadrillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1018" title="Orders of magnitude (numbers)">Quintillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1021" title="Orders of magnitude (numbers)">Sextillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1024" title="Orders of magnitude (numbers)">Septillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1027" title="Orders of magnitude (numbers)">Octillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1030" title="Orders of magnitude (numbers)">Nonillion</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)#1033" title="Orders of magnitude (numbers)">Decillion</a></li> <li><a href="/wiki/Eddington_number" title="Eddington number">Eddington number</a></li> <li><a href="/wiki/Googol" title="Googol">Googol</a></li> <li><a href="/wiki/Shannon_number" title="Shannon number">Shannon number</a></li> <li><a href="/wiki/Googolplex" title="Googolplex">Googolplex</a></li> <li><a href="/wiki/Skewes%27s_number" title="Skewes's number">Skewes's number</a></li> <li><a href="/wiki/Steinhaus%E2%80%93Moser_notation" title="Steinhaus–Moser notation">Moser's number</a></li> <li><a href="/wiki/Graham%27s_number" title="Graham's number">Graham's number</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">TREE(3)</a></li> <li><a href="/wiki/Friedman%27s_SSCG_function" title="Friedman's SSCG function">SSCG(3)</a></li> <li><a href="/wiki/Buchholz_hydra#BH(n)" title="Buchholz hydra">BH(3)</a></li> <li><a href="/wiki/Rayo%27s_number" title="Rayo's number">Rayo's number</a></li> <li><a class="mw-selflink selflink">Infinity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Expression<br />methods</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Notations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scientific_notation" title="Scientific notation">Scientific notation</a></li> <li><a href="/wiki/Knuth%27s_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow notation</a></li> <li><a href="/wiki/Conway_chained_arrow_notation" title="Conway chained arrow notation">Conway chained arrow notation</a></li> <li><a href="/wiki/Steinhaus%E2%80%93Moser_notation" title="Steinhaus–Moser notation">Steinhaus–Moser notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hyperoperation" title="Hyperoperation">Hyperoperation</a> <ul><li><a href="/wiki/Tetration" title="Tetration">Tetration</a></li> <li><a href="/wiki/Pentation" title="Pentation">Pentation</a></li></ul></li> <li><a href="/wiki/Ackermann_function" title="Ackermann function">Ackermann function</a></li> <li><a href="/wiki/Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a></li> <li><a href="/wiki/Fast-growing_hierarchy" title="Fast-growing hierarchy">Fast-growing hierarchy</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related <br />articles<br />(alphabetical <br />order)<br /></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Busy_beaver" title="Busy beaver">Busy beaver</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real number line</a></li> <li><a href="/wiki/Indefinite_and_fictitious_numbers" title="Indefinite and fictitious numbers">Indefinite and fictitious numbers</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known prime number</a></li> <li><a href="/wiki/List_of_numbers" title="List of numbers">List of numbers</a></li> <li><a href="/wiki/Long_and_short_scales" title="Long and short scales">Long and short scales</a></li> <li><a href="/wiki/Number" title="Number">Number systems</a></li> <li><a href="/wiki/Numeral_(linguistics)" title="Numeral (linguistics)">Number names</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)" title="Orders of magnitude (numbers)">Orders of magnitude</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of two</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of three</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Carl_Sagan#Sagan_units" title="Carl Sagan">Sagan Unit</a></li></ul> </div></td></tr><tr><td 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<script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-545c4fcd97-42f84","wgBackendResponseTime":193,"wgPageParseReport":{"limitreport":{"cputime":"1.057","walltime":"1.352","ppvisitednodes":{"value":6958,"limit":1000000},"postexpandincludesize":{"value":181229,"limit":2097152},"templateargumentsize":{"value":5212,"limit":2097152},"expansiondepth":{"value":23,"limit":100},"expensivefunctioncount":{"value":14,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":275998,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1090.583 1 -total"," 37.26% 406.343 1 Template:Reflist"," 11.83% 129.038 28 Template:Cite_book"," 11.66% 127.184 11 Template:Cite_web"," 8.08% 88.154 18 Template:Citation"," 7.52% 82.066 4 Template:Sister_project"," 7.35% 80.149 4 Template:Side_box"," 7.16% 78.080 1 Template:Unichar"," 7.10% 77.386 4 Template:Navbox"," 6.85% 74.687 1 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[\"CITEREFGemignani1990\"] = 1,\n [\"CITEREFGoodman1981\"] = 1,\n [\"CITEREFGosling2012\"] = 1,\n [\"CITEREFGowersBarrow-Green2008\"] = 1,\n [\"CITEREFGrattan-Guinness2005\"] = 1,\n [\"CITEREFGrime\"] = 1,\n [\"CITEREFHeathHeiberg1908\"] = 1,\n [\"CITEREFHutten1962\"] = 1,\n [\"CITEREFIan_Stewart2017\"] = 1,\n [\"CITEREFJain1982\"] = 1,\n [\"CITEREFJesseph1998\"] = 1,\n [\"CITEREFJoseph2000\"] = 1,\n [\"CITEREFKeisler1986\"] = 1,\n [\"CITEREFKline1972\"] = 1,\n [\"CITEREFKoupelisKuhn2007\"] = 1,\n [\"CITEREFMaddox2002\"] = 1,\n [\"CITEREFMarcus_Y._Yoo2011\"] = 1,\n [\"CITEREFMartin-Löf1990\"] = 1,\n [\"CITEREFMcLarty2014\"] = 1,\n [\"CITEREFMichael_FrameBenoit_Mandelbrot2002\"] = 1,\n [\"CITEREFMorris_Kline1985\"] = 1,\n [\"CITEREFO\u0026#039;Flaherty1986\"] = 1,\n [\"CITEREFRaoStetkær1991\"] = 1,\n [\"CITEREFRucker1995\"] = 1,\n [\"CITEREFRudolf_V_Rucker2019\"] = 1,\n [\"CITEREFRussell1996\"] = 1,\n [\"CITEREFSagan1994\"] = 1,\n [\"CITEREFSarton1928\"] = 1,\n [\"CITEREFSchattschneider2010\"] = 1,\n [\"CITEREFScott1981\"] = 1,\n [\"CITEREFSingh1988\"] = 1,\n [\"CITEREFStokes2012\"] = 1,\n [\"CITEREFSwokowski1983\"] = 1,\n [\"CITEREFTaylor1955\"] = 1,\n [\"CITEREFToker1989\"] = 1,\n [\"CITEREFWallace2004\"] = 1,\n [\"CITEREFWeeks2001\"] = 1,\n [\"CITEREFWeyl2012\"] = 1,\n [\"Chapter_4\"] = 1,\n}\ntemplate_list = table#1 {\n [\"About\"] = 1,\n [\"Analysis-footer\"] = 1,\n [\"Authority control\"] = 1,\n [\"Blockquote\"] = 2,\n [\"C.\"] = 2,\n [\"Citation\"] = 18,\n [\"Citation needed\"] = 2,\n [\"Cite IEP\"] = 1,\n [\"Cite SEP\"] = 1,\n [\"Cite arXiv\"] = 2,\n [\"Cite book\"] = 28,\n [\"Cite journal\"] = 7,\n [\"Cite news\"] = 1,\n [\"Cite web\"] = 11,\n [\"Commons category\"] = 1,\n [\"Distinguish\"] = 1,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Further\"] = 3,\n [\"Harvnb\"] = 11,\n [\"Harvtxt\"] = 1,\n [\"Hatnote group\"] = 1,\n [\"In Our Time\"] = 1,\n [\"Infinity\"] = 1,\n [\"Isbn\"] = 1,\n [\"Large numbers\"] = 1,\n [\"Main\"] = 4,\n [\"Math\"] = 7,\n [\"Pp-move\"] = 1,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 1,\n [\"See also\"] = 1,\n [\"Sfrac\"] = 2,\n [\"Short description\"] = 1,\n [\"Unichar\"] = 1,\n [\"Webarchive\"] = 9,\n [\"Wikibooks\"] = 1,\n [\"Wikiquote\"] = 1,\n [\"Wiktionary\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-5f6c89f4bf-7kt4k","timestamp":"20250401161421","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Infinity","url":"https:\/\/en.wikipedia.org\/wiki\/Infinity","sameAs":"http:\/\/www.wikidata.org\/entity\/Q205","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q205","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-12-30T07:33:10Z","dateModified":"2025-04-01T16:13:20Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/80\/SierpinskiTriangle.svg","headline":"mathematical concept"}</script> </body> </html>