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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">despintar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Entamu</div> </a> </li> <li id="toc-Teoría_de_conxuntos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teoría_de_conxuntos"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Teoría de conxuntos</span> </div> </a> <button aria-controls="toc-Teoría_de_conxuntos-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>Alternar subsección Teoría de conxuntos</span> </button> <ul id="toc-Teoría_de_conxuntos-sublist" class="vector-toc-list"> <li id="toc-Primer_definición_positiva_de_conxuntu_infinitu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primer_definición_positiva_de_conxuntu_infinitu"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Primer definición positiva de conxuntu infinitu</span> </div> </a> <ul id="toc-Primer_definición_positiva_de_conxuntu_infinitu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Númberos_ordinales_infinitos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Númberos_ordinales_infinitos"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Númberos ordinales infinitos</span> </div> </a> <ul id="toc-Númberos_ordinales_infinitos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primer_ordinal_infinitu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primer_ordinal_infinitu"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Primer ordinal infinitu</span> </div> </a> <ul id="toc-Primer_ordinal_infinitu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Númberos_cardinales_infinitos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Númberos_cardinales_infinitos"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Númberos cardinales infinitos</span> </div> </a> <ul id="toc-Númberos_cardinales_infinitos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analís_matemáticu" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analís_matemáticu"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Analís matemáticu</span> </div> </a> <ul id="toc-Analís_matemáticu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinitu_n'informática" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Infinitu_n'informática"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Infinitu n'informática</span> </div> </a> <ul id="toc-Infinitu_n'informática-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Historia</span> </div> </a> <button aria-controls="toc-Historia-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>Alternar subsección Historia</span> </button> <ul id="toc-Historia-sublist" class="vector-toc-list"> <li id="toc-El_símbolu_d'infinitu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#El_símbolu_d'infinitu"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>El símbolu d'infinitu</span> </div> </a> <ul id="toc-El_símbolu_d'infinitu-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ver_tamién" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_tamién"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ver tamién</span> </div> </a> <ul id="toc-Ver_tamién-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Referencies</span> </div> </a> <ul id="toc-Referencies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Más_información" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Más_información"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Más información</span> </div> </a> <ul id="toc-Más_información-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="Conteníu" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <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">Cambiar a la tabla de contenidos</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">Infinitu</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="Ir a un artículo en otro idioma. Disponible en 112 idiomas" > <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-112" 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">112 llingües</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 – alemán de Suiza" lang="gsw" hreflang="gsw" data-title="Unendlichkeit" data-language-autonym="Alemannisch" data-language-local-name="alemán de Suiza" 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 – amháricu" lang="am" hreflang="am" data-title="አዕላፍ" data-language-autonym="አማርኛ" data-language-local-name="amháricu" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Infinito" title="Infinito – aragonés" lang="an" hreflang="an" data-title="Infinito" data-language-autonym="Aragonés" data-language-local-name="aragonés" class="interlanguage-link-target"><span>Aragonés</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 – árabe" lang="ar" hreflang="ar" data-title="لانهاية" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</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 – árabe de Marruecos" lang="ary" hreflang="ary" data-title="لامسالية" data-language-autonym="الدارجة" data-language-local-name="árabe de Marruecos" class="interlanguage-link-target"><span>الدارجة</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 – árabe d’Exiptu" lang="arz" hreflang="arz" data-title="ملهاش نهايه" data-language-autonym="مصرى" data-language-local-name="árabe d’Exiptu" class="interlanguage-link-target"><span>مصرى</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 – asamés" lang="as" hreflang="as" data-title="অসীম" data-language-autonym="অসমীয়া" data-language-local-name="asamés" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sonsuzluq" title="Sonsuzluq – azerbaixanu" lang="az" hreflang="az" data-title="Sonsuzluq" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaixanu" 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-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-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Begal%C4%ABb%C4%97" title="Begalībė – samogitianu" lang="sgs" hreflang="sgs" data-title="Begalībė" data-language-autonym="Žemaitėška" data-language-local-name="samogitianu" class="interlanguage-link-target"><span>Žemaitėška</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 – bielorrusu" lang="be" hreflang="be" data-title="Бесканечнасць" data-language-autonym="Беларуская" data-language-local-name="bielorrusu" 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 – búlgaru" lang="bg" hreflang="bg" data-title="Безкрайност" data-language-autonym="Български" data-language-local-name="búlgaru" class="interlanguage-link-target"><span>Български</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-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 – bengalín" lang="bn" hreflang="bn" data-title="অসীম" data-language-autonym="বাংলা" data-language-local-name="bengalín" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Beskona%C4%8Dnost" title="Beskonačnost – bosniu" lang="bs" hreflang="bs" data-title="Beskonačnost" data-language-autonym="Bosanski" data-language-local-name="bosniu" 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 – catalán" lang="ca" hreflang="ca" data-title="Infinit" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</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 – kurdu central" lang="ckb" hreflang="ckb" data-title="بێکۆتایی" data-language-autonym="کوردی" data-language-local-name="kurdu central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Infinitu" title="Infinitu – corsu" lang="co" hreflang="co" data-title="Infinitu" data-language-autonym="Corsu" data-language-local-name="corsu" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Nekone%C4%8Dno" title="Nekonečno – checu" lang="cs" hreflang="cs" data-title="Nekonečno" data-language-autonym="Čeština" data-language-local-name="checu" class="interlanguage-link-target"><span>Čeština</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-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Anfeidredd" title="Anfeidredd – galés" lang="cy" hreflang="cy" data-title="Anfeidredd" data-language-autonym="Cymraeg" data-language-local-name="galés" 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 – danés" lang="da" hreflang="da" data-title="Uendelighed" data-language-autonym="Dansk" data-language-local-name="danés" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Unendlich_(Mathematik)" title="Unendlich (Mathematik) – alemán" lang="de" hreflang="de" data-title="Unendlich (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</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 – griegu" lang="el" hreflang="el" data-title="Άπειρο" data-language-autonym="Ελληνικά" data-language-local-name="griegu" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Infinity" title="Infinity – inglés" lang="en" hreflang="en" data-title="Infinity" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Infinito" title="Infinito – español" lang="es" hreflang="es" data-title="Infinito" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/L%C3%B5pmatus" title="Lõpmatus – estoniu" lang="et" hreflang="et" data-title="Lõpmatus" data-language-autonym="Eesti" data-language-local-name="estoniu" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Infinitu" title="Infinitu – vascu" lang="eu" hreflang="eu" data-title="Infinitu" data-language-autonym="Euskara" data-language-local-name="vascu" 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 – persa" lang="fa" hreflang="fa" data-title="بی‌نهایت" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</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 – finlandés" lang="fi" hreflang="fi" data-title="Äärettömyys" data-language-autonym="Suomi" data-language-local-name="finlandés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Infini" title="Infini – francés" lang="fr" hreflang="fr" data-title="Infini" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/%C3%9Cnentelkhaid" title="Ünentelkhaid – frisón del norte" lang="frr" hreflang="frr" data-title="Ünentelkhaid" data-language-autonym="Nordfriisk" data-language-local-name="frisón del norte" class="interlanguage-link-target"><span>Nordfriisk</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 – irlandés" lang="ga" hreflang="ga" data-title="Éigríoch" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</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 – chinu gan" lang="gan" hreflang="gan" data-title="無限" data-language-autonym="贛語" data-language-local-name="chinu gan" class="interlanguage-link-target"><span>贛語</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-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Infinito" title="Infinito – gallegu" lang="gl" hreflang="gl" data-title="Infinito" data-language-autonym="Galego" data-language-local-name="gallegu" class="interlanguage-link-target"><span>Galego</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 – guyaratí" lang="gu" hreflang="gu" data-title="અનંત" data-language-autonym="ગુજરાતી" data-language-local-name="guyaratí" class="interlanguage-link-target"><span>ગુજરાતી</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 – hebréu" lang="he" hreflang="he" data-title="אינסוף" data-language-autonym="עברית" data-language-local-name="hebréu" class="interlanguage-link-target"><span>עברית</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-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Anant" title="Anant – hindi de Fiji" lang="hif" hreflang="hif" data-title="Anant" data-language-autonym="Fiji Hindi" data-language-local-name="hindi de Fiji" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Beskona%C4%8Dnost" title="Beskonačnost – croata" lang="hr" hreflang="hr" data-title="Beskonačnost" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/V%C3%A9gtelen" title="Végtelen – húngaru" lang="hu" hreflang="hu" data-title="Végtelen" data-language-autonym="Magyar" data-language-local-name="húngaru" class="interlanguage-link-target"><span>Magyar</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) – armeniu" lang="hy" hreflang="hy" data-title="Անվերջություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeniu" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Takhingga" title="Takhingga – indonesiu" lang="id" hreflang="id" data-title="Takhingga" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiu" class="interlanguage-link-target"><span>Bahasa Indonesia</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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%93endanleiki" title="Óendanleiki – islandés" lang="is" hreflang="is" data-title="Óendanleiki" data-language-autonym="Íslenska" data-language-local-name="islandés" 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) – italianu" lang="it" hreflang="it" data-title="Infinito (matematica)" data-language-autonym="Italiano" data-language-local-name="italianu" class="interlanguage-link-target"><span>Italiano</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 – xaponés" lang="ja" hreflang="ja" data-title="無限" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Infiniti" title="Infiniti – inglés criollu xamaicanu" lang="jam" hreflang="jam" data-title="Infiniti" data-language-autonym="Patois" data-language-local-name="inglés criollu xamaicanu" class="interlanguage-link-target"><span>Patois</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-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 – xeorxanu" lang="ka" hreflang="ka" data-title="უსასრულობა" data-language-autonym="ქართული" data-language-local-name="xeorxanu" 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 – kazaquistanín" lang="kk" hreflang="kk" data-title="Шексіздік" data-language-autonym="Қазақша" data-language-local-name="kazaquistanín" class="interlanguage-link-target"><span>Қазақша</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 – canarés" lang="kn" hreflang="kn" data-title="ಅನಂತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarés" class="interlanguage-link-target"><span>ಕನ್ನಡ</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 – coreanu" lang="ko" hreflang="ko" data-title="무한" data-language-autonym="한국어" data-language-local-name="coreanu" class="interlanguage-link-target"><span>한국어</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 – curdu" lang="ku" hreflang="ku" data-title="Bêdawî" data-language-autonym="Kurdî" data-language-local-name="curdu" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Didhiwedhter" title="Didhiwedhter – córnicu" lang="kw" hreflang="kw" data-title="Didhiwedhter" data-language-autonym="Kernowek" data-language-local-name="córnicu" class="interlanguage-link-target"><span>Kernowek</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 – kirguistanín" lang="ky" hreflang="ky" data-title="Чексиздик" data-language-autonym="Кыргызча" data-language-local-name="kirguistanín" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="artículos destacaos"><a href="https://la.wikipedia.org/wiki/Infinitas" title="Infinitas – llatín" lang="la" hreflang="la" data-title="Infinitas" data-language-autonym="Latina" data-language-local-name="llatín" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Begalyb%C4%97" title="Begalybė – lituanu" lang="lt" hreflang="lt" data-title="Begalybė" data-language-autonym="Lietuvių" data-language-local-name="lituanu" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bezgal%C4%ABba" title="Bezgalība – letón" lang="lv" hreflang="lv" data-title="Bezgalība" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Tsiefa" title="Tsiefa – malgaxe" lang="mg" hreflang="mg" data-title="Tsiefa" data-language-autonym="Malagasy" data-language-local-name="malgaxe" class="interlanguage-link-target"><span>Malagasy</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 – macedoniu" lang="mk" hreflang="mk" data-title="Бесконечност" data-language-autonym="Македонски" data-language-local-name="macedoniu" class="interlanguage-link-target"><span>Македонски</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-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 – mongol" lang="mn" hreflang="mn" data-title="Хязгааргүй" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</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-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ketakterhinggaan" title="Ketakterhinggaan – malayu" lang="ms" hreflang="ms" data-title="Ketakterhinggaan" data-language-autonym="Bahasa Melayu" data-language-local-name="malayu" class="interlanguage-link-target"><span>Bahasa Melayu</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 – birmanu" lang="my" hreflang="my" data-title="အနန္တ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmanu" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Unendlichkeid" title="Unendlichkeid – baxu alemán" lang="nds" hreflang="nds" data-title="Unendlichkeid" data-language-autonym="Plattdüütsch" data-language-local-name="baxu alemán" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Oneindigheid" title="Oneindigheid – neerlandés" lang="nl" hreflang="nl" data-title="Oneindigheid" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Uendeleg" title="Uendeleg – noruegu Nynorsk" lang="nn" hreflang="nn" data-title="Uendeleg" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegu Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Uendelig" title="Uendelig – noruegu Bokmål" lang="nb" hreflang="nb" data-title="Uendelig" data-language-autonym="Norsk bokmål" data-language-local-name="noruegu Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Infinit" title="Infinit – occitanu" lang="oc" hreflang="oc" data-title="Infinit" data-language-autonym="Occitan" data-language-local-name="occitanu" class="interlanguage-link-target"><span>Occitan</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 – punyabí" lang="pa" hreflang="pa" data-title="ਅਨੰਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punyabí" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</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 – polacu" lang="pl" hreflang="pl" data-title="Nieskończoność" data-language-autonym="Polski" data-language-local-name="polacu" class="interlanguage-link-target"><span>Polski</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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Infinito" title="Infinito – portugués" lang="pt" hreflang="pt" data-title="Infinito" data-language-autonym="Português" data-language-local-name="portugués" 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 – rumanu" lang="ro" hreflang="ro" data-title="Infinit" data-language-autonym="Română" data-language-local-name="rumanu" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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 – rusu" lang="ru" hreflang="ru" data-title="Бесконечность" data-language-autonym="Русский" data-language-local-name="rusu" class="interlanguage-link-target"><span>Русский</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-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Nfinitu_(matim%C3%A0tica)" title="Nfinitu (matimàtica) – sicilianu" lang="scn" hreflang="scn" data-title="Nfinitu (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="sicilianu" class="interlanguage-link-target"><span>Sicilianu</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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Beskona%C4%8Dnost_(matematika)" title="Beskonačnost (matematika) – serbo-croata" lang="sh" hreflang="sh" data-title="Beskonačnost (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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 – cingalés" lang="si" hreflang="si" data-title="අනන්තය" data-language-autonym="සිංහල" data-language-local-name="cingalés" 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 – eslovacu" lang="sk" hreflang="sk" data-title="Nekonečno" data-language-autonym="Slovenčina" data-language-local-name="eslovacu" 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 – eslovenu" lang="sl" hreflang="sl" data-title="Neskončnost" data-language-autonym="Slovenščina" data-language-local-name="eslovenu" class="interlanguage-link-target"><span>Slovenščina</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-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Pafund%C3%ABsia" title="Pafundësia – albanu" lang="sq" hreflang="sq" data-title="Pafundësia" data-language-autonym="Shqip" data-language-local-name="albanu" class="interlanguage-link-target"><span>Shqip</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 – serbiu" lang="sr" hreflang="sr" data-title="Бесконачност" data-language-autonym="Српски / srpski" data-language-local-name="serbiu" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/O%C3%A4ndlighet" title="Oändlighet – suecu" lang="sv" hreflang="sv" data-title="Oändlighet" data-language-autonym="Svenska" data-language-local-name="suecu" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%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-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 – taxiquistanín" lang="tg" hreflang="tg" data-title="Беинтиҳоӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="taxiquistanín" class="interlanguage-link-target"><span>Тоҷикӣ</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 – tailandés" lang="th" hreflang="th" data-title="อนันต์" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sonsuz" title="Sonsuz – turcu" lang="tr" hreflang="tr" data-title="Sonsuz" data-language-autonym="Türkçe" data-language-local-name="turcu" class="interlanguage-link-target"><span>Türkçe</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 – tártaru" lang="tt" hreflang="tt" data-title="Чиксезлек" data-language-autonym="Татарча / tatarça" data-language-local-name="tártaru" class="interlanguage-link-target"><span>Татарча / tatarça</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 – ucraín" lang="uk" hreflang="uk" data-title="Нескінченність" data-language-autonym="Українська" data-language-local-name="ucraín" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Cheksizlik" title="Cheksizlik – uzbequistanín" lang="uz" hreflang="uz" data-title="Cheksizlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbequistanín" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Lopm%C3%A4tomuz" title="Lopmätomuz – vepsiu" lang="vep" hreflang="vep" data-title="Lopmätomuz" data-language-autonym="Vepsän kel’" data-language-local-name="vepsiu" 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 – vietnamín" lang="vi" hreflang="vi" data-title="Vô tận" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamín" class="interlanguage-link-target"><span>Tiếng Việt</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 – chinu wu" lang="wuu" hreflang="wuu" data-title="无穷" data-language-autonym="吴语" data-language-local-name="chinu 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 mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%97%A0%E7%A9%B7" title="无穷 – chinu" lang="zh" hreflang="zh" data-title="无穷" data-language-autonym="中文" data-language-local-name="chinu" class="interlanguage-link-target"><span>中文</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 – chinu lliterariu" lang="lzh" hreflang="lzh" data-title="無限" data-language-autonym="文言" data-language-local-name="chinu lliterariu" class="interlanguage-link-target"><span>文言</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 – chinu min nan" lang="nan" hreflang="nan" data-title="Bû-hān" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="chinu min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</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 – cantonés" lang="yue" hreflang="yue" data-title="無窮盡" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q205#sitelinks-wikipedia" title="Editar los enllaces d'interllingua" class="wbc-editpage">Editar los enllaces</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacios de nome"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Infinitu" title="Ver la páxina de conteníu [c]" accesskey="c"><span>Páxina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Alderique:Infinitu&action=edit&redlink=1" rel="discussion" class="new" title="Alderique tocante al conteníu de la páxina (la páxina nun esiste) [t]" accesskey="t"><span>Alderique</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambiar variante de idioma" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">asturianu</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistes"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Infinitu"><span>Lleer</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Infinitu&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Infinitu&action=edit" title="Editar el códigu fonte d'esta páxina [e]" accesskey="e"><span>Editar la fonte</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Infinitu&action=history" title="Versiones antigües d'esta páxina [h]" accesskey="h"><span>Ver historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramientes de páxina"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramientes" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Ferramientes</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Ferramientes</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">despintar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Más opciones" > <div class="vector-menu-heading"> Aiciones </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Infinitu"><span>Lleer</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Infinitu&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Infinitu&action=edit" title="Editar el códigu fonte d'esta páxina [e]" accesskey="e"><span>Editar la fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Infinitu&action=history"><span>Ver historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Xeneral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:LoQueEnlazaAqu%C3%AD/Infinitu" title="Llista de toles páxines wiki qu'enllacien equí [j]" accesskey="j"><span>Lo qu'enllaza equí</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:CambiosEnEnlazadas/Infinitu" rel="nofollow" title="Cambios recientes nes páxines enllazaes dende esta [k]" accesskey="k"><span>Cambios rellacionaos</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=ast" title="Xubir ficheros [u]" accesskey="u"><span>Xubir ficheru</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasEspeciales" title="Llista de toles páxines especiales [q]" accesskey="q"><span>Páxines especiales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Infinitu&oldid=4271211" title="Enllaz permanente a esta revisión de la páxina"><span>Enllaz permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Infinitu&action=info" title="Más información sobro esta páxina"><span>Información de la páxina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Infinitu&id=4271211&wpFormIdentifier=titleform" title="Información tocante a cómo citar esta páxina"><span>Citar esta páxina</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:Acortador_de_URL&url=https%3A%2F%2Fast.wikipedia.org%2Fwiki%2FInfinitu"><span>Llograr la URL encurtiada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fast.wikipedia.org%2Fwiki%2FInfinitu"><span>Xenerar códigu QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprentar/esportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libro&bookcmd=book_creator&referer=Infinitu"><span>Crear un llibru</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Infinitu&action=show-download-screen"><span>Descargar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Infinitu&printable=yes" title="Versión imprentable d'esta páxina [p]" accesskey="p"><span>Versión pa imprentar</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> N'otros proyeutos </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Infinity" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q205" title="Enllaz al elementu del depósitu de datos coneutáu [g]" accesskey="g"><span>Elementu de Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Ferramientes de páxina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">despintar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-tradubot" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="https://es.wikipedia.org/wiki/Infinito" title="Esti artículu foi traducíu automáticamente y precisa revisase manualmente"><img alt="Esti artículu foi traducíu automáticamente y precisa revisase manualmente" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/16px-Robot_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/24px-Robot_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/32px-Robot_icon.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">De Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ast" dir="ltr"><p><br /> </p> <table class="infobox plantia-xenerica" style="font-size:90%;width:25em"><tbody><tr><th colspan="2" style="text-align:center;font-size:125%;font-weight:bold;background-color: #acacac">Infinitu</th></tr><tr><td colspan="2" style="text-align:center;background-color: #cdcdcd"> conceutu matemáticu</td></tr><tr><td colspan="2" style="text-align:center;background-color: #cdcdcd"> cardinalidad <sup>(es)</sup> <span class="mw-valign-baseline skin-invert" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q4049983?uselang=ast" title="Traducir"><img alt="Traducir" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/10px-Noun_Project_label_icon_1116097_cc_mirror.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/15px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/20px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 2x" data-file-width="158" data-file-height="161" /></a></span></td></tr><tr><td colspan="2" style="text-align:center"> <span typeof="mw:File"><a href="/wiki/Ficheru:Infinite.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Infinite.svg/260px-Infinite.svg.png" decoding="async" width="260" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Infinite.svg/390px-Infinite.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Infinite.svg/520px-Infinite.svg.png 2x" data-file-width="330" data-file-height="220" /></a></span></td></tr><tr><td colspan="2" style="text-align:right"><span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q205" title="Cambiar los datos en Wikidata"><img alt="Cambiar los datos en Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/12px-Arbcom_ru_editing.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/18px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/24px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></td></tr></tbody></table> <p>El conceutu de <b>infinitu</b> apaez en delles cañes de la <a href="/wiki/Matem%C3%A1tica" class="mw-redirect" title="Matemática">matemática</a>, la <a href="/wiki/Filosof%C3%ADa" title="Filosofía">filosofía</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> y l'<a href="/wiki/Astronom%C3%ADa" title="Astronomía">astronomía</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> en referencia a una cantidá ensin llende o final, contrapuestu al conceutu de <a href="https://ast.wiktionary.org/wiki/finito" class="extiw" title="wikt:finito">finitud</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>En matématicas l'infinitu apaez de diverses formes: en <a href="/wiki/Xeometr%C3%ADa" title="Xeometría">xeometría</a>, el <a href="/w/index.php?title=Puntu_al_infinitu&action=edit&redlink=1" class="new" title="Puntu al infinitu (la páxina nun esiste)">puntu al infinitu</a> en <a href="/w/index.php?title=Xeometr%C3%ADa_proyectiva&action=edit&redlink=1" class="new" title="Xeometría proyectiva (la páxina nun esiste)">xeometría proyectiva</a> y el <a href="/wiki/Puntu_de_fuga" title="Puntu de fuga">puntu de fuga</a> en <a href="/w/index.php?title=Xeometr%C3%ADa_descriptiva&action=edit&redlink=1" class="new" title="Xeometría descriptiva (la páxina nun esiste)">xeometría descriptiva</a>; n'analís matemáticu, los <a href="/w/index.php?title=Llende_d%27una_funci%C3%B3n_llendes&action=edit&redlink=1" class="new" title="Llende d'una función llendes (la páxina nun esiste)">Llende d'una función llendes</a> infinitos, o <a href="/w/index.php?title=Llendes_al_infinitu&action=edit&redlink=1" class="new" title="Llendes al infinitu (la páxina nun esiste)">llendes al infinitu</a>; y en <a href="/wiki/Teor%C3%ADa_de_conxuntos" title="Teoría de conxuntos">teoría de conxuntos</a> como <a href="/w/index.php?title=Transfinito&action=edit&redlink=1" class="new" title="Transfinito (la páxina nun esiste)">númberos transfinitos</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Teoría_de_conxuntos"><span id="Teor.C3.ADa_de_conxuntos"></span>Teoría de conxuntos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=1" title="Editar seición: Teoría de conxuntos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=1" title="Editar el código fuente de la sección: Teoría de conxuntos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Los conxuntos finitos tienen una propiedá "intuitiva" que los caracteriza; dada una parte mesma de los mesmos, ésta contién un númberu d'elementos menor que tol conxuntu. Esto ye, nun puede establecese una biyección ente una parte mesma del conxuntu finito y tol conxuntu. Sicasí, esa propiedá "intuitiva" de los conxuntos finitos nun la tienen los conxuntos infinitos, y formalmente dicimos que: </p> <dl><dd><i>Un conxuntu <span class="mwe-math-element"><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> <mspace width="thickmathspace" /> </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/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span> ye infinitu si esiste un subconxuntu propiu <span class="mwe-math-element"><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> <mspace width="thickmathspace" /> </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/bd396c58c818f98c8e1c42f41ab81729df9e407b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.409ex; height:2.176ex;" alt="{\displaystyle B\;}"></span> de <span class="mwe-math-element"><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> <mspace width="thickmathspace" /> </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/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span>, esto ye, un subconxuntu <span class="mwe-math-element"><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\subset A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊂</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subset A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670e1f664373a6eb64b063d1856ddc49a527366e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle B\subset A}"></span> tal que <span class="mwe-math-element"><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\neq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≠</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\neq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78362703472ea51edc4614b6b7a7bda8e83131c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle A\neq B}"></span>, tal qu'esiste una biyección <span class="mwe-math-element"><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:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\to B}"></span> ente <span class="mwe-math-element"><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> <mspace width="thickmathspace" /> </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/f61491be9bf07fe6ba295510dc36c210469fe1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle A\;}"></span> y <span class="mwe-math-element"><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> <mspace width="thickmathspace" /> </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/bd396c58c818f98c8e1c42f41ab81729df9e407b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.409ex; height:2.176ex;" alt="{\displaystyle B\;}"></span>.</i></dd></dl> <p>La idea de <a href="/w/index.php?title=Cardinalidad&action=edit&redlink=1" class="new" title="Cardinalidad (la páxina nun esiste)">cardinalidad</a> d'un <a href="/wiki/Conxuntu" title="Conxuntu">conxuntu</a> basar na noción anterior de <a href="/w/index.php?title=Biyecci%C3%B3n&action=edit&redlink=1" class="new" title="Biyección (la páxina nun esiste)">biyección</a>. De dos conxuntos ente los que puede establecese una biyección dizse que tienen la mesma cardinalidad. Pa un conxuntu finito el so cardinalidad puede representase por un <a href="/wiki/N%C3%BAmberu_natural" title="Númberu natural">númberu natural</a>. Por exemplu, el conxuntu <i>{mazana, pera, durazno}</i> tien 3 elementos. Esto significa de manera más formal que puede establecese una <a href="/w/index.php?title=Biyecci%C3%B3n&action=edit&redlink=1" class="new" title="Biyección (la páxina nun esiste)">biyección</a> ente tal conxuntu y el númberu 3 que ye'l conxuntu {0,1,2}: </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 {\begin{matrix}{\mbox{Mazana}}&\leftrightarrow &0\\{\mbox{Pera}}&\leftrightarrow &1\\{\mbox{Durazno}}&\leftrightarrow &2\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Mazana</mtext> </mstyle> </mrow> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Pera</mtext> </mstyle> </mrow> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Durazno</mtext> </mstyle> </mrow> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\mbox{Mazana}}&\leftrightarrow &0\\{\mbox{Pera}}&\leftrightarrow &1\\{\mbox{Durazno}}&\leftrightarrow &2\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdf97c225c2888065e4da4d5213679e24707657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.512ex; height:9.176ex;" alt="{\displaystyle {\begin{matrix}{\mbox{Mazana}}&\leftrightarrow &0\\{\mbox{Pera}}&\leftrightarrow &1\\{\mbox{Durazno}}&\leftrightarrow &2\end{matrix}}}"></span></dd></dl> <p>Dichu d'otra forma, ye posible faer pareyes (0, mazana), (1, pera), (2, durazno) de cuenta que cada elementu de los dos conxuntos utilícese esautamente una vegada. Cuando ye posible establecer tal rellación "unu a unu" ente dos conxuntos dizse que dambos conxuntos tienen <i>la mesma cardinalidad</i>, lo cual, pa conxuntos finitos, equival a que tengan el mesmu númberu d'elementos. </p> <div class="mw-heading mw-heading3"><h3 id="Primer_definición_positiva_de_conxuntu_infinitu"><span id="Primer_definici.C3.B3n_positiva_de_conxuntu_infinitu"></span>Primer definición positiva de conxuntu infinitu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=2" title="Editar seición: Primer definición positiva de conxuntu infinitu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=2" title="Editar el código fuente de la sección: Primer definición positiva de conxuntu infinitu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La primer definición positiva de conxuntu infinitu foi dada por <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> y básase na siguiente observación: Si un conxuntu <i>S</i> ye finito y <i>T</i> ye un <a href="/w/index.php?title=Subconxuntu&action=edit&redlink=1" class="new" title="Subconxuntu (la páxina nun esiste)">subconxuntu propiu</a>, nun ye posible construyir una biyección ente <i>S</i> y <i>T</i>. Por exemplu, si <i>S</i> = {1,2,3,4,5,6,7,8} y <i>T</i> = {2,4,6,8} nun ye posible construyir una biyección ente <i>S</i> y <i>T</i>, porque de ser asina tendríen la mesma cardinalidad (el mesmu númberu d'elementos). </p><p>Un conxuntu ye infinitu si ye posible atopar un subconxuntu propiu del mesmu que tenga la mesma cardinalidad que'l conxuntu orixinal. Consideremos el conxuntu de los númberos naturales <i>N</i>={1,2,3,4,5,...}, que ye un conxuntu infinitu. Pa verificar tal afirmación ye necesariu atopar un subconxuntu propiu y construyir una biyección ente dambos. Pa esti casu, consideremos el conxuntu d'enteros positivos pares <i>P</i>={2,4,6,8,10,...}. El conxuntu <i>P</i> ye un subconxuntu propiu de <i>N</i>, y la regla de asignación <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/116da78ed2740d907954709f88129fe361cdffaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.566ex; height:2.176ex;" alt="{\displaystyle n\to 2n}"></span> ye una biyección: </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 \mathbf {A} ={\begin{bmatrix}N&\leftrightarrow &P\\1&\leftrightarrow &2\\2&\leftrightarrow &4\\3&\leftrightarrow &6\\4&\leftrightarrow &8\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>N</mi> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mi>P</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mo stretchy="false">↔</mo> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}N&\leftrightarrow &P\\1&\leftrightarrow &2\\2&\leftrightarrow &4\\3&\leftrightarrow &6\\4&\leftrightarrow &8\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd632e961d111db96e7cbef86d9ec27208ec8d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:20.395ex; height:15.843ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}N&\leftrightarrow &P\\1&\leftrightarrow &2\\2&\leftrightarrow &4\\3&\leftrightarrow &6\\4&\leftrightarrow &8\end{bmatrix}}.}"></span></dd></dl> <p>yá que a tou elementu de <i>N</i> correspuénde-y un únicu elementu de <i>P</i> y viceversa. </p> <div class="mw-heading mw-heading3"><h3 id="Númberos_ordinales_infinitos"><span id="N.C3.BAmberos_ordinales_infinitos"></span>Númberos ordinales infinitos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=3" title="Editar seición: Númberos ordinales infinitos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=3" title="Editar el código fuente de la sección: Númberos ordinales infinitos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r4219085">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/w/index.php?title=N%C3%BAmberu_ordinal_(teor%C3%ADa_de_conxuntos)&action=edit&redlink=1" class="new" title="Númberu ordinal (teoría de conxuntos) (la páxina nun esiste)">Númberu ordinal (teoría de conxuntos)</a></div> <p>Los númberos ordinales sirven pa notar una posición nun <a href="/w/index.php?title=Conxuntu_orden%C3%A1u&action=edit&redlink=1" class="new" title="Conxuntu ordenáu (la páxina nun esiste)">conxuntu ordenáu</a> (primer, segundu, tercer elementu ...). L'exemplu más elemental ye'l de los <a href="/wiki/N%C3%BAmberos_naturales" class="mw-redirect" title="Númberos naturales">númberos naturales</a>, que se definen rigorosamente asina: Nótase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4b06f9315849466a0502680377e30a9da8a1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 0\,}"></span> el conxuntu vacíu: </p> <style data-mw-deduplicate="TemplateStyles:r4219090">.mw-parser-output .ecuacion{padding:5px 10px;background-color:var(--background-color-base);color:var(--color-base);margin-left:30px;margin-bottom:0.8em;margin-top:0.5em;min-width:50%}.mw-parser-output .ecuacion .referencia{float:right;width:10%;text-align:end}.mw-parser-output .ecuacion cite{font-style:normal}</style><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\{\}=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\{\}=\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc3a27821f80965147b0d4f7ca5a8d47e8a6d5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.492ex; height:2.843ex;" alt="{\displaystyle 0=\{\}=\varnothing }"></span> </p> </blockquote> <p>nótase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 1\,}"></span> el conxuntu que namái contién <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4b06f9315849466a0502680377e30a9da8a1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 0\,}"></span>: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=\{0\}=\{\varnothing \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=\{0\}=\{\varnothing \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2839c46a6de2d00a77c771e244537010149a7d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.98ex; height:2.843ex;" alt="{\displaystyle 1=\{0\}=\{\varnothing \}}"></span> </p> </blockquote> <p>depués nótase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f0585b3d3c0d207a91af7a41e4173b58f309ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 2\,}"></span> el conxuntu que namái contién <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4b06f9315849466a0502680377e30a9da8a1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 0\,}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 1\,}"></span>: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2=\{0,1\}=\{0,\{0\}\}=\{\varnothing ,\{\varnothing \}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2=\{0,1\}=\{0,\{0\}\}=\{\varnothing ,\{\varnothing \}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39531396dd3dd3d8460a995de5e6777bd871eff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.45ex; height:2.843ex;" alt="{\displaystyle 2=\{0,1\}=\{0,\{0\}\}=\{\varnothing ,\{\varnothing \}\}}"></span> </p> </blockquote> <p>Y asina socesivamente: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3=\{0,1,2\}=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\},\qquad (n+1)=n\bigcup \{n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>⋃</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3=\{0,1,2\}=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\},\qquad (n+1)=n\bigcup \{n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/770aab7efcaf800d78a53e98bdbdedd0961f3663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:58.941ex; height:3.843ex;" alt="{\displaystyle 3=\{0,1,2\}=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\},\qquad (n+1)=n\bigcup \{n\}}"></span> </p> </blockquote> <p>Por construcción, 0 ta incluyíu en 1, quién de la mesma ta incluyíu en 2, yá que obviamente: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\subseteq n\bigcup \{n\}=(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⊆</mo> <mi>n</mi> <mo>⋃</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\subseteq n\bigcup \{n\}=(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/980802f2493ba7e8842a09e7e48b8a6028da2b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.882ex; height:3.843ex;" alt="{\displaystyle n\subseteq n\bigcup \{n\}=(n+1)}"></span> </p> </blockquote> <p>La inclusión dexa convertir a los ordinales nun <a href="/w/index.php?title=Conxuntu_bien_orden%C3%A1u&action=edit&redlink=1" class="new" title="Conxuntu bien ordenáu (la páxina nun esiste)">conxuntu bien ordenáu</a> (dos elementos distintos siempres pueden comparase, y añadiendo la igualdá daría un orde total) ente estos conxuntos que se prefier, por costume, escribir "<", lo que da les rellaciones 0 < 1 < 2 < 3. Dicir qu'un ordinal ye menor (puramente) qu'otru significa, cuando se-yos considera a dambos como conxuntos, que ta incluyíu nel otru. </p><p>Si <i>a</i> y <i>b</i> son ordinales, entós <i>a</i>O<i>b</i>, la <a href="/w/index.php?title=Uni%C3%B3n_de_conxuntos&action=edit&redlink=1" class="new" title="Unión de conxuntos (la páxina nun esiste)">unión de los conxuntos</a>, tamién ye un ordinal. En particular, si son ordinales finitos (conxuntos finitos) correspondientes a los naturales <i>a</i> y <i>b</i>, entós <i>a</i>O<i>b</i> correspuende al mayor de los dos, <i>a</i> o <i>b</i>. Polo xeneral, si los conxuntos <i>a<sub>i</sub></i> son ordinales, onde <i>i</i> toma tolos valores d'un conxuntu <i>I</i>, entós <i>a</i> = O<i>a<sub>i</sub></i> tamién lo será. Y si el conxuntu <i>I</i> nun ye finito, tampoco lo será <i>a</i>. Asina vamos llograr ordinales (esto ye númberos) infinitos. </p><p>Acabamos de cayer nuna "trampa", al falar de conxuntu finito ensin definir el conceutu. Pa definilo rigorosamente, tenemos de comparalo colos ordinales. Dos conxuntos bien ordenaos <i>A</i> y <i>B</i> son isomorfos (con rellación al orde) si esiste una <a href="/w/index.php?title=Funci%C3%B3n_biyectiva&action=edit&redlink=1" class="new" title="Función biyectiva (la páxina nun esiste)">biyección</a> <i>f</i> ente dambos que respeta l'orde: si <i>a</i> < <i>a'</i> en <i>A</i>, entós <i>f</i>(<i>a</i>) < <i>f</i>(<i>a</i>) en <i>B</i>. Resulta obviu constatar que si <i>A</i> ye un conxuntu ordenáu con <i>n</i> elementos (<i>n</i> enteru natural) entós <i>A</i> ye isomorfu <i>a<sub>n</sub></i> = {0, 1, 2, ..., n-1}. Basta con renombrar cada elementu de <i>A</i> pa llograr <i>A</i> = {<i>a</i><sub>0</sub>, <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ..., <i>a</i><sub><i>n</i>-1</sub>}. Un isomorfismu ye puramente un cambéu d'apelación. Vamos Dicir qu'un ordinal ye finito si caúna de les sos partes non vacíes tien un elementu máximu. Polo tanto tou natural ye un ordenal finito. La intuición diznos que nun hai otros ordenales finitos. Lóxicamente, vamos dicir qu'un conxuntu ordenáu ye finito si ye <a href="/wiki/Isomorfismu" title="Isomorfismu">isomorfu</a> a un ordinal finito, esto ye a un natural. </p><p>Pa introducir los ordinales infinitos, ye precisu dar agora la definición exacta d'un ordinal: </p> <dl><dd><i>Un conxuntu A totalmente ordenáu (pola inclusión) ye un ordinal si y namái si cada elementu d'A ye tamién un subconxuntu d'A</i></dd></dl> <p>Yá vimos que ye'l casu pa los naturales: Por exemplu, el conxuntu 2 = {0, 1} almite 1= {0}, como elementu y polo tanto tamién como subconxuntu. </p><p><b>Tou conxuntu bien ordenáu ye isomorfu a un ordinal.</b> Esto ye obviu nel casu finito, y amuésase por inducción transfinita que lo ye nel casu infinitu. Esto ye, renombrando los elementos d'un conxuntu bien ordenáu siempres llogramos un ordinal. </p> <div class="mw-heading mw-heading3"><h3 id="Primer_ordinal_infinitu">Primer ordinal infinitu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=4" title="Editar seición: Primer ordinal infinitu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=4" title="Editar el código fuente de la sección: Primer ordinal infinitu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Yá vimos qu'una unión cualesquier d'ordinales ye un ordinal. Si tomamos una unión finita d'ordinales finitos, fabricamos un ordinal finito. Pa llograr el primer ordinal infinitu tenemos qu'axuntar un númberu non finito d'ordinales finitos. Faciéndolo, siempres cayemos nel mesmu conxuntu, construyíu al axuntar tolos ordinales finitos, ye dicir los naturales. El conxuntu de tolos naturales, ℕ, ye pos el primer ordinal infinitu, lo que nun tendría de sorprender, y notar nesti contestu ω (omega). </p><p>Pa visualizar los ordinales, resulta bien práuticu representar cada unu por un puntu d'una socesión creciente converxente, como por casu o<sub>n</sub> = 1 - 1/(n+1). Esto da daqué asemeyáu a: </p> <dl><dd>X__________X_________X_______X______X______X_____X____X___X__X_X_XXX........</dd></dl> <p>Escoyamos un puntu de la socesión, y miremos cuantos puntos tán más a la izquierda. Nel exemplu, hai cuatro, y polo tanto tratar d'o<sub>4</sub>, lo que correspuende al ordinal 4. Pa representar l'ordinal w, resulta natural añader a la socesión previa un puntu 'O' asitiáu esautamente na llende de la socesión: </p> <dl><dd>X__________X_________X_______X______X______X_____X____X___X__X_X_XXX...O</dd></dl> <p>A la izquierda d'o<sub>w</sub> hai una infinidá de puntos, polo tanto w ye infinitu. Pero si escoyemos a cualesquier otru puntu de la socesión a la so esquierda, yá nun ye'l casu, lo cual prueba que w ye'l primer ordinal infinitu. Dempués de w llega w+1, w+2 ... que se representen añadiendo a la derecha unu dos o más puntos, primeramente distantes, y depués más cercanos ente sigo: </p> <dl><dd>X________X________X_______X______X______X_____X____X___X__X_X_XXX...O_______X_____X</dd></dl> <p>L'últimu puntu dibuxáu correspuende a w+2. </p><p>Más xeneralmente, pa sumar dos ordinales A y B camúdense los nomes de los elementos por que sían toos distintos, depués xúntense los conxuntos A y B, poniendo B a la derecha d'A ye dicir imponiendo que cada elementu de B seya mayor que tolos d'A. Asina construyimos w+1, ... y asina podemos construyir 1+w: Notemos Y el elementu de 1, y X los de w: </p> <dl><dd>Y__________X__________X_________X_______X______X______X_____X____X___X__X_X_XXX...</dd></dl> <p>Salta a la vista que w y 1+w son bien paecíos. De fechu la función x →x - 1 realiza un isomorfismu ente ellos (1+w tien dos elementos llamaos 0: 0<sub>A</sub> y 0<sub>B</sub>. El primeru fai'l papel de -1 na función). Polo tanto correspuenden al mesmu ordinal: 1+w = w. Mas nun ye'l casu de w+1, que ye distintu de w porque'l so el conxuntu w+1 tien un <a href="/w/index.php?title=Elementu_m%C3%A1ximu&action=edit&redlink=1" class="new" title="Elementu máximu (la páxina nun esiste)">elementu máximu</a> (l'O del dibuxu) ente que'l conxuntu w nun lo tien (la llende de los naturales nun ye un natural). </p><p>El puntu w (l'O del dibuxu) nun tien antecesor, ye dicir que nun esiste un n tal que n+1=w: dizse que w ye una ordinal llende. Cero tien tamién esta propiedá pero nun merez esta apelación. Como w+1 ≠ 1+w, la adición nun ye conmutativa nos ordinales. </p><p>Constrúyese de la mesma w + w que se nota lóxicamente 2w. La multiplicación definir a partir de la adición como pa los naturales. </p><p>Una vegada que se representó nw, con n natural, nun resulta demasiáu difícil imaxinar lo que va ser w.w, escritu w². Depués puede definise w<sup>n</sup>, con n natural, y, tomando la llende, w<sup>w</sup>, tien tantos elementos como la recta real. </p><p>La socesión <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\omega ^{{\dots }^{\omega }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo>…</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\omega ^{{\dots }^{\omega }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a43017317e154ab7ce00bbb431d055cabd10807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.391ex; height:3.176ex;" alt="{\displaystyle \omega ^{\omega ^{{\dots }^{\omega }}}}"></span> tien como llende <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb0a8377db20e42274444cb181d51b5532b5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Númberos_cardinales_infinitos"><span id="N.C3.BAmberos_cardinales_infinitos"></span>Númberos cardinales infinitos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=5" title="Editar seición: Númberos cardinales infinitos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=5" title="Editar el código fuente de la sección: Númberos cardinales infinitos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/w/index.php?title=N%C3%BAmberu_cardinal_(teor%C3%ADa_de_conxuntos)&action=edit&redlink=1" class="new" title="Númberu cardinal (teoría de conxuntos) (la páxina nun esiste)">Númberu cardinal (teoría de conxuntos)</a></div> <p>El cardinal d'un conxuntu ye'l númberu d'elementos que contién. Esta noción ye polo tanto distinta del ordinal, que caracteriza'l llugar d'un elementu nuna socesión. <i>"Cinco"</i> difier de <i>"quintu"</i> anque obviamente esiste una rellación ente dambos. Dizse que dos conxuntos tienen el mesmu cardinal si esiste una biyección ente ellos. Contrariamente a los ordinales, esta biyección nun tien que respetar l'orde (amás los conxuntos nun tienen que ser ordenaos). </p><p>Como yá tenemos un surtíu de conxuntos -los ordinales- veamos los sos tamaños (esto ye los sos cardinales) respeutivos. Nun ye nenguna sorpresa que los ordinales finitos tamién son cardinales: ente dos conxuntos con n y m elementos, m y n distintos, nun puede haber biyección, polo tanto tienen cardinales distintos. Pero nun ye'l casu colos ordenales infinitos: Por exemplu, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64eefa705e5e12f34fc2fe138f4897cd4980f374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.449ex; height:2.343ex;" alt="{\displaystyle \omega +1}"></span> tán en biyección pola función: </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 \omega +1\to \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">→</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega +1\to \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a308e6a6f427d79c1ae82a6c1c636cafc308494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.509ex; height:2.343ex;" alt="{\displaystyle \omega +1\to \omega }"></span></dd> <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 x\to x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e70da5240f4049380a32ad2f5cd6b6e8500820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.276ex; height:2.343ex;" alt="{\displaystyle x\to x+1}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4241175143df08751786f77b63440654ff3b55b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.222ex; height:2.176ex;" alt="{\displaystyle \omega \to 0}"></span>, tal biyección nun respeta l'orde, por eso dos ordinales distintos pueden corresponder a un mesmu cardinal.</dd></dl> <p>Suelse notar |A| el cardinal d'A. Llámase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span> (<i>alef<sub>0</sub></i>) el cardinal de w, esto ye del conxuntu de los naturales (onde <i>alef</i> ye la primer lletra del alfabetu hebréu). </p><p>Si A y B son conxuntos, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle |A\times B|=|A|\cdot |B|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo>×</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle |A\times B|=|A|\cdot |B|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f4fae9fbc882e5523ff88372f11292bec4733e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.719ex; height:2.176ex;" alt="{\displaystyle \scriptstyle |A\times B|=|A|\cdot |B|}"></span>, onde x designa'l <a href="/w/index.php?title=Productu_cartesianu&action=edit&redlink=1" class="new" title="Productu cartesianu (la páxina nun esiste)">productu cartesianu</a> de los conxuntos, y "·" ye'l productu de los cardinales definíos por esta fórmula. El conxuntu de les partes d'un conxuntu A, P(A) ta en biyección col conxuntu de les funciones d'A escontra {0,1}, conxuntu que d'escribe 2<sup>A</sup>, como casu particular de Y<sup>X</sup> que denota el conxuntu de les aplicaciones de X escontra Y. </p><p>El cardinal de <b>R</b>, conxuntu de los reales, ye polo tanto 2<sup>alef<sub>0</sub></sup>, porque <b>R</b> ta en biyección coles partes de <b>N</b>, per mediu de la escritura decimal de los reales. </p><p>Nun puede decidise, colos axomes clásicos (los de la teoría de los conxuntos, fundamentos de la matemática), si esiste un cardinal mayor que alef<sub>0</sub> y menor que 2<sup>alef<sub>0</sub></sup>, ye dicir si esiste un conxuntu con más elementos que <b>N</b> pero con menos elementos que <b>R</b>. La <i>hipótesis del continuu</i>, que ye un axoma adicional, afirma que non. </p> <div class="mw-heading mw-heading2"><h2 id="Analís_matemáticu"><span id="Anal.C3.ADs_matem.C3.A1ticu"></span>Analís matemáticu</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=6" title="Editar seición: Analís matemáticu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=6" title="Editar el código fuente de la sección: Analís matemáticu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un conxuntu de númberos reales <i>S</i> ye <a href="/w/index.php?title=Acut%C3%A1u&action=edit&redlink=1" class="new" title="Acutáu (la páxina nun esiste)">acutáu</a> superiormente si esiste un númberu <i>c</i> (la cota) tal que <i>c</i> ye mayor que tou elementu de <i>S</i> (Por exemplu, si <i>S</i>={π ; 7 ; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43595011bdeb48a850f90f103d754fc62d03e5bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.485ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\,\!}"></span>} entós <i>S</i> ye un conxuntu acutáu, una y bones el númberu <i>c</i>=10 cumple que π<10, 7<10, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43595011bdeb48a850f90f103d754fc62d03e5bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.485ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\,\!}"></span><10). Cuando un conxuntu nun ye acutáu, pa cualquier númberu <i>c</i> ye posible atopar <span class="mwe-math-element"><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\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51186ba8afb2067573a9082d55dd383df1ea9214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle x\in S}"></span> de cuenta que <i>c < x</i>. El conceutu d'infinitu introduzse como una cota especial pa esti tipu de conxuntos. Esti conceutu d'infinitu representar col símbolu <span class="mwe-math-element"><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>. </p><p>Tamién ye utilizáu nel <a href="/wiki/Anal%C3%ADs_matem%C3%A1ticu" title="Analís matemáticu">analís matemáticu</a> cuando quier espresase que los términos d'una <a href="/wiki/Socesi%C3%B3n_matem%C3%A1tica" title="Socesión matemática">socesión ordenada</a>, o los valores que toma una función al tomar la variable dependiente valores cercanos a unu fitu primeramente "diverxe" ("tiende a infinitu", o'l so <a href="/wiki/Llende_matem%C3%A1tica" title="Llende matemática">llende</a> ye infinitu). Nesti contestu, considérase <span class="mwe-math-element"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/4735c5366157d0969eb244dec5a007a3ec6da38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.711ex; height:1.676ex;" alt="{\displaystyle \infty \,\!}"></span> pa representar a la llende que tiende a infinitu y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fcd4157907e63b2975f620e5259bebe0636662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 0\,\!}"></span> a la llende cuando tiende a 0; y non al númberu <a href="/wiki/Cero" title="Cero">0</a>). </p><p>Pa recordar les regles de llende suelse entós allegar a les siguientes regles <a href="/w/index.php?title=Nemotecnia&action=edit&redlink=1" class="new" title="Nemotecnia (la páxina nun esiste)">nemotecnies</a>: (equí "x" representa un n° real cualesquier) </p> <ul><li><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 x+\infty =\infty \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+\infty =\infty \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13ff9caf2cee6d08e3a897d33cb49fc6368a89e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:12.303ex; height:2.176ex;" alt="{\displaystyle x+\infty =\infty \,\!}"></span> y <span class="mwe-math-element"><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+(-\infty )=(-\infty )\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+(-\infty )=(-\infty )\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2a6f025c38824a51c0008e9631c713024f921a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:19.538ex; height:2.843ex;" alt="{\displaystyle x+(-\infty )=(-\infty )\,\!}"></span></dd> <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 x-\infty =-\infty \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-\infty =-\infty \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d5a5d751ee77d60e5ab071a6fa32c587f06a19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:14.111ex; height:2.176ex;" alt="{\displaystyle x-\infty =-\infty \,\!}"></span></dd> <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 x-(-\infty )=\infty \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-(-\infty )=\infty \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caf4396da051248d3cc8038f7478725820ddd3cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:15.921ex; height:2.843ex;" alt="{\displaystyle x-(-\infty )=\infty \,\!}"></span></dd> <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 {x \over \infty }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi mathvariant="normal">∞</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x \over \infty }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985bf30301bfe3d33cf3cecec9963b65a339b283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.421ex; height:4.676ex;" alt="{\displaystyle {x \over \infty }=0}"></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 {x \over -\infty }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x \over -\infty }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab02e1b2592fc6173101849b41be8c3d80d8cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.229ex; height:4.843ex;" alt="{\displaystyle {x \over -\infty }=0}"></span></dd> <dd>Si <span class="mwe-math-element"><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>0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29e66161d850cc06163cc3ada197e8d9c321fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.978ex; height:2.176ex;" alt="{\displaystyle x>0\,\!}"></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 x\cdot \infty =\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot \infty =\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960838a5db2f066607e0c62eec8e010996c3def6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.755ex; height:1.676ex;" alt="{\displaystyle x\cdot \infty =\infty }"></span>   y   <span class="mwe-math-element"><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\cdot (-\infty )=(-\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot (-\infty )=(-\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb54566478bba1e913f6ba1b370e8561ff39c149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.99ex; height:2.843ex;" alt="{\displaystyle x\cdot (-\infty )=(-\infty )}"></span>.</dd> <dd>Si <span class="mwe-math-element"><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<0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eadf00afe7960d8e00a51d6b9e8ccb890425cb28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.978ex; height:2.176ex;" alt="{\displaystyle x<0\,\!}"></span>   entós   <span class="mwe-math-element"><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\cdot \infty =-\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot \infty =-\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da0bedcc164654ba4f2bc36278e29dc7c6eafee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.563ex; height:2.176ex;" alt="{\displaystyle x\cdot \infty =-\infty }"></span> y <span class="mwe-math-element"><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\cdot (-\infty )=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot (-\infty )=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32931a54631485c3f303bab090f7f4c8f37ca5e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.372ex; height:2.843ex;" alt="{\displaystyle x\cdot (-\infty )=\infty }"></span>.</dd></dl></li></ul> <ul><li><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 \infty +\infty =\infty ,\qquad (-\infty )+(-\infty )=-\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty +\infty =\infty ,\qquad (-\infty )+(-\infty )=-\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b99774e5059b585af713a96fe9b8337b4161c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.542ex; height:2.843ex;" alt="{\displaystyle \infty +\infty =\infty ,\qquad (-\infty )+(-\infty )=-\infty }"></span></dd> <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 \infty \cdot \infty =\infty ,\qquad (-\infty )(-\infty )=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mo>⋅</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty \cdot \infty =\infty ,\qquad (-\infty )(-\infty )=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc51962d2764f648371b5f5878e978664bd6727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.733ex; height:2.843ex;" alt="{\displaystyle \infty \cdot \infty =\infty ,\qquad (-\infty )(-\infty )=\infty }"></span></dd></dl></li></ul> <p>Llendes indeterminaes (nun ye posible determinar <a href="/w/index.php?title=A_priori&action=edit&redlink=1" class="new" title="A priori (la páxina nun esiste)">a priori</a> el so valor como nel restu de los exemplos, nun hai un valor asignáu): </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 0\cdot \infty \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot \infty \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e80877604c550fba11d050cf81b38db55884a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.552ex; height:2.176ex;" alt="{\displaystyle 0\cdot \infty \,}"></span></dd> <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 0\cdot (-\infty )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot (-\infty )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d584d1c252f352db3c64aab0e67e287e8f9de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.17ex; height:2.843ex;" alt="{\displaystyle 0\cdot (-\infty )\,}"></span></dd> <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 \infty +(-\infty )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty +(-\infty )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2715584306324bb496a2c98245da2c059204c841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.492ex; height:2.843ex;" alt="{\displaystyle \infty +(-\infty )\,}"></span></dd> <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 \infty -\infty \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty -\infty \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6328ba9c837992782fece04e3c61c178d643cf1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.875ex; height:2.176ex;" alt="{\displaystyle \infty -\infty \,}"></span></dd> <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 {\pm \infty \over \pm \infty }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>±</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow> <mo>±</mo> <mi mathvariant="normal">∞</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\pm \infty \over \pm \infty }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64102d55efdca3327be72119309a8e3e6e005004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.355ex; height:5.176ex;" alt="{\displaystyle {\pm \infty \over \pm \infty }\,}"></span></dd> <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 {(\pm \infty )}^{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>±</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(\pm \infty )}^{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7587d64d616052a1182bc5690e8fd0ece6f534e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.383ex; height:3.343ex;" alt="{\displaystyle {(\pm \infty )}^{0}\,}"></span></dd> <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 1^{\pm \infty }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{\pm \infty }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0b838296a35eb1eb4922454059a83abafd5ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.703ex; height:2.676ex;" alt="{\displaystyle 1^{\pm \infty }\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Infinitu_n'informática"><span id="Infinitu_n.27inform.C3.A1tica"></span>Infinitu n'informática</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=7" title="Editar seición: Infinitu n'informática" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=7" title="Editar el código fuente de la sección: Infinitu n'informática"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De manera rellacionada col infinitu pa númberos reales, dalgunos <a href="/wiki/Llinguaxe_de_programaci%C3%B3n" title="Llinguaxe de programación">llinguaxes de programación</a> almiten un valor especial que recibe'l nome de <i>infinitu</i>: valor que puede llograse como resultáu de ciertes operaciones matemátiques non realizables, tales como les descrites nel puntu anterior o operaciones teóricamente posibles, pero demasiáu complexes pal so trabayu nel ordenador/llinguaxe en cuestión. N'otros llinguaxes a cencielles produciríase un error. </p> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=8" title="Editar seición: Historia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=8" title="Editar el código fuente de la sección: Historia"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="El_símbolu_d'infinitu"><span id="El_s.C3.ADmbolu_d.27infinitu"></span>El símbolu d'infinitu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=9" title="Editar seición: El símbolu d'infinitu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=9" title="Editar el código fuente de la sección: El símbolu d'infinitu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:John_Wallis_by_Sir_Godfrey_Kneller,_Bt.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/John_Wallis_by_Sir_Godfrey_Kneller%2C_Bt.jpg/180px-John_Wallis_by_Sir_Godfrey_Kneller%2C_Bt.jpg" decoding="async" width="180" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/John_Wallis_by_Sir_Godfrey_Kneller%2C_Bt.jpg/270px-John_Wallis_by_Sir_Godfrey_Kneller%2C_Bt.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/John_Wallis_by_Sir_Godfrey_Kneller%2C_Bt.jpg/360px-John_Wallis_by_Sir_Godfrey_Kneller%2C_Bt.jpg 2x" data-file-width="2400" data-file-height="2900" /></a><figcaption><a href="/w/index.php?title=John_Wallis&action=edit&redlink=1" class="new" title="John Wallis (la páxina nun esiste)">John Wallis</a> foi'l primer matemáticu n'usar el símbolu d'infinitu nes sos obres.</figcaption></figure> <p>Los oríxenes del símbolu d'infinitu <span class="mwe-math-element"><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> son inciertos. Puesto que la forma asemeyar a la curva <a href="/w/index.php?title=Lemniscata&action=edit&redlink=1" class="new" title="Lemniscata (la páxina nun esiste)">lemniscata</a> (del llatín <i>lemniscus</i>, ye dicir <i>cinta</i>), suxirióse que representa un llazu zarráu. </p><p>Tamién se cree posible que la forma provenga d'otros símbolos alquímicos o relixosos, como por casu ciertes representaciones de la culiebra <a href="/w/index.php?title=Ur%C3%B3boros&action=edit&redlink=1" class="new" title="Uróboros (la páxina nun esiste)">uróboros</a>. El matemáticu <a href="/w/index.php?title=John_Wallis&action=edit&redlink=1" class="new" title="John Wallis (la páxina nun esiste)">John Wallis</a> foi'l primeru n'usar el símbolu <span class="mwe-math-element"><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> pa representar al infinitu nel so tratáu <i>De sectionibus conicus</i> en <a href="/wiki/1655" title="1655">1655</a>. </p><p>Quíxose ver tamién una <a href="/w/index.php?title=Banda_de_M%C3%B6bius&action=edit&redlink=1" class="new" title="Banda de Möbius (la páxina nun esiste)">banda de Möbius</a> na so forma, anque'l símbolu usar mientres cientos d'años primero que <a href="/w/index.php?title=August_M%C3%B6bius&action=edit&redlink=1" class="new" title="August Möbius (la páxina nun esiste)">August Möbius</a> afayara la banda que lleva'l so nome. </p><p>El símbolu d'infinitu representar en <a href="/wiki/Unicode" title="Unicode">Unicode</a> col calter <font size="+1">∞</font> (U+221E). </p> <table class="wikitable" border="1" cellspacing="1" cellpadding="5" align="center"> <caption><b>Cronoloxía</b><sup id="cite_ref-50COSES_4-0" class="reference"><a href="#cite_note-50COSES-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr bgcolor="#e1ecf7"> <th>Añu </th> <th>Acontecimientu </th></tr> <tr align="left" bgcolor="#f0f5fa"> <td>350 <a href="/wiki/E.C." class="mw-redirect" title="E.C.">e.C.</a> </td> <td><a href="/wiki/Arist%C3%B3teles" title="Aristóteles">Aristóteles</a> refuga un infinitu real. </td></tr> <tr align="left" bgcolor="#f0f5fa"> <td><a href="/wiki/1639" title="1639">1639</a> </td> <td><a href="/w/index.php?title=G%C3%A9rard_Desargues&action=edit&redlink=1" class="new" title="Gérard Desargues (la páxina nun esiste)">Gérard Desargues</a> introduz la idea del infinitu na <a href="/wiki/Xeometr%C3%ADa" title="Xeometría">xeometría</a>. </td></tr> <tr align="left" bgcolor="#f0f5fa"> <td><a href="/wiki/1655" title="1655">1655</a> </td> <td>Atribuyir a <a href="/w/index.php?title=John_Wallis&action=edit&redlink=1" class="new" title="John Wallis (la páxina nun esiste)">John Wallis</a> ser el primeru n'utilizar el<br />símbolu <span class="mwe-math-element"><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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </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/1eff5c079e87ea939725600fea5268f23d53f0cf" 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> pal infinitu. </td></tr> <tr align="left" bgcolor="#f0f5fa"> <td><a href="/wiki/1874" title="1874">1874</a> </td> <td><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> especifíca, na teoría de conxuntos, distintos<br /> órdenes d'infinitu. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Ver_tamién"><span id="Ver_tami.C3.A9n"></span>Ver tamién</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=10" title="Editar seición: Ver tamién" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=10" title="Editar el código fuente de la sección: Ver tamién"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=As%C3%ADntota&action=edit&redlink=1" class="new" title="Asíntota (la páxina nun esiste)">Asíntota</a></li> <li><a href="/w/index.php?title=Conxuntu_numerable&action=edit&redlink=1" class="new" title="Conxuntu numerable (la páxina nun esiste)">Conxuntos numerables</a></li> <li><a href="/w/index.php?title=Infinitesimal&action=edit&redlink=1" class="new" title="Infinitesimal (la páxina nun esiste)">Infinitesimal</a></li> <li><a href="/w/index.php?title=N%C3%BAmberu_transfinito&action=edit&redlink=1" class="new" title="Númberu transfinito (la páxina nun esiste)">Númberu transfinito</a></li> <li><a href="/w/index.php?title=Paradoja&action=edit&redlink=1" class="new" title="Paradoja (la páxina nun esiste)">Paradoxes sobre l'infinitu</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencies">Referencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=11" title="Editar seición: Referencies" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=11" title="Editar el código fuente de la sección: Referencies"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3503771">@media only screen and (max-width:600px){.mw-parser-output .llistaref{column-count:1!important}}</style><div class="llistaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite style="font-style:normal" id="Referencia-Monnoyeur-1995">Monnoyeur, Francoise (1995). <i>L'infinitu de los matemáticos, l'infinitu de los filósofos (Infini des mathématiciens, infini des philosophes)</i>. Paris: Belin. <a href="/wiki/Especial:FuentesDeLibros/978-2701110189" title="Especial:FuentesDeLibros/978-2701110189">ISBN 978-2701110189</a>.</cite></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite style="font-style:normal" id="Referencia-Monnoyeur-1999">Monnoyeur, Francoise (1999). <i>L'Infinitu de los filósofos, l'infinitu de los astrónomos (Infini des philosophes, infini des astronomes)</i>. Paris: Belin. <a href="/wiki/Especial:FuentesDeLibros/978-2701115207" title="Especial:FuentesDeLibros/978-2701115207">ISBN 978-2701115207</a>.</cite></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation cita-Journal" id="CITAREFFedriani2010">Fedriani, Eugenio M. (2010). «<a rel="nofollow" class="external text" href="http://www.fisem.org/web/union/revistes/21/Union_021_008.pdf">Matemátiques del más allá: l'infinitu</a>». <i>Unión: Revista Iberoamericana d'Educación Matemática</i> <b>21</b>:  p. 37-58. <small><a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/1815-0640">1815-0640</a></small><span class="printonly">. <a rel="nofollow" class="external free" href="http://www.fisem.org/web/union/revistes/21/Union_021_008.pdf">http://www.fisem.org/web/union/revistes/21/Union_021_008.pdf</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Matem%C3%A1tiques+del+m%C3%A1s+all%C3%A1%3A+l%27infinitu&rft.jtitle=Uni%C3%B3n%3A+Revista+Iberoamericana+d%27Educaci%C3%B3n+Matem%C3%A1tica&rft.aulast=Fedriani&rft.aufirst=Eugenio+M.&rft.au=Fedriani%2C%26%2332%3BEugenio+M.&rft.date=2010&rft.volume=21&rft.pages=%26nbsp%3Bp.%26nbsp%3B37-58&rft.issn=1815-0640&rft_id=http%3A%2F%2Fwww.fisem.org%2Fweb%2Funion%2Frevistes%2F21%2FUnion_021_008.pdf&rfr_id=info:sid/ast.wikipedia.org:Infinitu"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-50COSES-4"><span class="mw-cite-backlink"><a href="#cite_ref-50COSES_4-0">↑</a></span> <span class="reference-text"><cite style="font-style:normal">Tony Crilly (2011). <i>50 coses qu'hai que saber sobre matemátiques</i>. Ed. Ariel. <a href="/wiki/Especial:FuentesDeLibros/9789871496099" class="internal mw-magiclink-isbn">ISBN 978-987-1496-09-9</a>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Más_información"><span id="M.C3.A1s_informaci.C3.B3n"></span>Más información</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Infinitu&veaction=edit&section=12" title="Editar seición: Más información" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Infinitu&action=edit&section=12" title="Editar el código fuente de la sección: Más información"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Manolios, Panagiotis & Vroon, Daron. Algorithms for ordinal arithmetic. Baader, Franz (ed), 19th International Conference on Automated Deduction--CADE-19. Pages 243-257 of LNAI, vol. 2741. Springer-Verlag.</li></ul> <p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r2260362">.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid #c8ccd1;background-color:#eaecf0;color:#222222}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}</style><div class="mw-authority-control navigation-not-searchable"><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r4075543">.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 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.navbar{float:left;text-align:left;margin-right:0.5em}</style></div><div role="navigation" class="navbox" aria-labelledby="Control_d&#039;autoridaes" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_d&#039;autoridaes" scope="row" class="navbox-group" style="width:1%;width: 12%; text-align:center;"><a href="/wiki/Ayuda:Control_d%27autoridaes" title="Ayuda:Control d'autoridaes">Control d'autoridaes</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><b>Proyeutos Wikimedia</b></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q205" class="extiw" title="wikidata:Q205">Q205</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Infinity">Infinity</a></span></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikiquote" title="Wikiquote"><img alt="Wikiquote" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/15px-Wikiquote-logo.svg.png" decoding="async" width="15" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/30px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span> Cites famoses:</span> <span class="uid"><a href="https://ast.wikiquote.org/wiki/Infinito" class="extiw" title="q:Infinito">Infinito</a></span></li></ul> <hr /> <ul><li><b>Identificadores</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Integrated_Authority_File" class="mw-redirect" title="Integrated Authority File">GND</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4136067-9">4136067-9</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/National_Diet_Library" class="mw-redirect" title="National Diet Library">NDL</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00576691">00576691</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Art_%26_Architecture_Thesaurus" title="Art & Architecture Thesaurus">AAT</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.getty.edu/vow/AATFullDisplay?find=&logic=AND&note=&subjectid=300073212">300073212</a></span></li> <li><b>Diccionarios y enciclopedies</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Enciclopedia_Brit%C3%A1nica" class="mw-redirect" title="Enciclopedia Británica">Britannica</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/infinity-mathematics">url</a></span></li></ul> </div></td></tr></tbody></table></div><div class="mw-mf-linked-projects hlist"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q205" class="extiw" title="wikidata:Q205">Q205</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Infinity">Infinity</a></span></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikiquote" title="Wikiquote"><img alt="Wikiquote" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/15px-Wikiquote-logo.svg.png" decoding="async" width="15" height="18" class="mw-file-element" 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