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</div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Аксиоматска теорија скупова</span></h1> <div class="tagline"></div> </div> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Језик" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> <span class="minerva-icon minerva-icon--language"></span> <span>Језик</span> </a> </li> <li id="page-actions-watch" class="page-actions-menu__list-item"> <a role="button" id="ca-watch" 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cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Уреди</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="sr" dir="ltr"><section class="mf-section-0" id="mf-section-0"><p>У <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика">математици</a>, <b>аксиоматска теорија скупова</b> је ригорозна реформулација <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D1%81%D0%BA%D1%83%D0%BF%D0%BE%D0%B2%D0%B0" title="Теорија скупова">теорије скупова</a> у <a href="/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0_%D0%BF%D1%80%D0%B2%D0%BE%D0%B3_%D1%80%D0%B5%D0%B4%D0%B0" title="Логика првог реда">логици првог реда</a> настала да реши парадоксе <a href="/wiki/Naivna_teorija_skupova" title="Naivna teorija skupova">наивне теорије скупова</a>. Основу теорије скупова је у највећој мери развио немачки математичар <a href="/wiki/%D0%93%D0%B5%D0%BE%D1%80%D0%B3_%D0%9A%D0%B0%D0%BD%D1%82%D0%BE%D1%80" title="Георг Кантор">Георг Кантор</a> крајем 19. века. </p><p>У почетку контроверзна, теорија скупова је добила улогу <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%B5_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B5" title="Основе математике">основне теорије</a> у модерној математици, у смислу да интерпретира идеје о математичким објектима (бројевима, функцијама...) из свих традиционалних области математике (<a href="/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Алгебра">алгебра</a>, <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Математичка анализа">анализа</a>, <a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%98%D0%B0" title="Топологија">топологија</a>...) у једној теорији, и пружа стандардан скуп аксиома за њихово доказивање или оповргавање. У исто време, основни концепти теорије скупова се користе у целој математици. Треба такође напоменути да постој ематематичари који користе и промовишу другачије приступе основи математике. </p><p>Основни концепти теорије скупова су скуп и припадност скупу. <a href="/wiki/%D0%A1%D0%BA%D1%83%D0%BF" title="Скуп">Скупом</a> се сматра свака колекција објеката, који се називају <i>члановима</i> (или <i>елементима</i>) скупа. У математици чланови скупова су било који математички објекти, који и сами могу бити скупови. Тако на пример имамо скуп <b>N</b> <a href="/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Природан број">природних бројева</a> {0, 1, 2, 3, 4, ...}, скуп <a href="/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реалан број">реалних бројева</a>, као и скуп <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функција (математика)">функција</a> које скуп природних бројева пресликавају у скуп природних бројева; али на пример и скуп {0, 2, <b>N</b>} чији су чланови бројеви 0 и 2 и скуп <b>N</b>. </p><p>У почетку је развијено оно шта се данас анзива <a href="/wiki/Naivna_teorija_skupova" title="Naivna teorija skupova">наивном теоријом скупова</a>. Испоставило се да ако се узме да се свака дата операција може обављати на било ком скупу без рестрикција, долази до <a href="/w/index.php?title=%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81%D0%B8_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B5_%D1%81%D0%BA%D1%83%D0%BF%D0%BE%D0%B2%D0%B0&action=edit&redlink=1" class="new" title="Парадокси теорије скупова (страница не постоји)">парадокса</a> попут <a href="/wiki/%D0%A0%D0%B0%D1%81%D0%B5%D0%BB%D0%BE%D0%B2_%D0%BF%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81" title="Раселов парадокс">Раселовог парадокса</a>. Да би се ови проблеми решили, теорија скупова је реконструисана, коришћењем <a href="/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0" title="Аксиома">аксиоматског</a> приступа. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Види_још"><span id=".D0.92.D0.B8.D0.B4.D0.B8_.D1.98.D0.BE.D1.88"></span>Види још</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D1%82%D1%81%D0%BA%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D1%81%D0%BA%D1%83%D0%BF%D0%BE%D0%B2%D0%B0&action=edit&section=1" title="Уредите одељак „Види још”" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>уреди</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <ul><li><a href="/w/index.php?title=%D0%90%D0%BB%D1%82%D0%B5%D1%80%D0%BD%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D1%81%D0%BA%D1%83%D0%BF%D0%BE%D0%B2%D0%B0&action=edit&redlink=1" class="new" title="Алтернативна теорија скупова (страница не постоји)">Алтернативна теорија скупова</a></li> <li><a href="/w/index.php?title=%D0%91%D0%B5%D1%82_%D0%B1%D1%80%D0%BE%D1%98&action=edit&redlink=1" class="new" title="Бет број (страница не постоји)">Бет број</a></li> <li><a href="/w/index.php?title=%D0%9A%D0%B0%D0%BD%D1%82%D0%BE%D1%80-%D0%91%D0%B5%D1%80%D0%BD%D1%88%D1%82%D0%B0%D1%98%D0%BD-%D0%A8%D1%80%D0%B5%D0%B4%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0&action=edit&redlink=1" class="new" title="Кантор-Бернштајн-Шредерова теорема (страница не постоји)">Кантор-Бернштајн-Шредерова теорема</a></li> <li><a href="/w/index.php?title=%D0%9A%D0%B0%D0%BD%D1%82%D0%BE%D1%80%D0%BE%D0%B2_%D0%B4%D0%B8%D1%98%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%B8_%D0%BF%D0%BE%D1%81%D1%82%D1%83%D0%BF%D0%B0%D0%BA&action=edit&redlink=1" class="new" title="Канторов дијагонални поступак (страница не постоји)">Канторов дијагонални поступак</a></li> <li><a href="/wiki/Kantorova_teorema" title="Kantorova teorema">Канторова теорема</a></li> <li><a href="/wiki/Naivna_teorija_skupova" title="Naivna teorija skupova">Наивна теорија скупова</a></li> <li><a href="/wiki/%D0%97%D0%B5%D1%80%D0%BC%D0%B5%D0%BB%D0%BE-%D0%A4%D1%80%D0%B5%D0%BD%D0%BA%D0%B5%D0%BB_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D1%81%D0%BA%D1%83%D0%BF%D0%BE%D0%B2%D0%B0" class="mw-redirect" title="Зермело-Френкел теорија скупова">Зермело-Френкел теорија скупова</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r25469611">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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href="https://it.wikipedia.org/wiki/Teoria_assiomatica_degli_insiemi" title="Teoria assiomatica degli insiemi — италијански" lang="it" hreflang="it" data-title="Teoria assiomatica degli insiemi" data-language-autonym="Italiano" data-language-local-name="италијански" 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/%E5%85%AC%E7%90%86%E7%9A%84%E9%9B%86%E5%90%88%E8%AB%96" title="公理的集合論 — јапански" lang="ja" hreflang="ja" data-title="公理的集合論" data-language-autonym="日本語" data-language-local-name="јапански" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Axiomatikus_halmazelm%C3%A9let" title="Axiomatikus halmazelmélet — мађарски" lang="hu" hreflang="hu" data-title="Axiomatikus halmazelmélet" data-language-autonym="Magyar" data-language-local-name="мађарски" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Axiomatische_verzamelingenleer" title="Axiomatische verzamelingenleer — холандски" lang="nl" hreflang="nl" data-title="Axiomatische verzamelingenleer" data-language-autonym="Nederlands" data-language-local-name="холандски" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2" title="Аксиоматическая теория множеств — руски" lang="ru" hreflang="ru" data-title="Аксиоматическая теория множеств" data-language-autonym="Русский" data-language-local-name="руски" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aksiomaattinen_joukko-oppi" title="Aksiomaattinen joukko-oppi — фински" lang="fi" hreflang="fi" data-title="Aksiomaattinen joukko-oppi" data-language-autonym="Suomi" data-language-local-name="фински" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%A4ngdteoretiska_axiom" title="Mängdteoretiska axiom — шведски" lang="sv" hreflang="sv" data-title="Mängdteoretiska axiom" data-language-autonym="Svenska" data-language-local-name="шведски" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D1%96%D0%BE%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%97_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD" title="Аксіоматика теорії множин — украјински" lang="uk" hreflang="uk" data-title="Аксіоматика теорії множин" data-language-autonym="Українська" data-language-local-name="украјински" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%85%AC%E7%90%86%E5%8C%96%E9%9B%86%E5%90%88%E8%AE%BA" title="公理化集合论 — кинески" lang="zh" hreflang="zh" data-title="公理化集合论" data-language-autonym="中文" data-language-local-name="кинески" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AD%90%E9%9B%86%E5%85%AC%E7%90%86" title="子集公理 — кантонски" lang="yue" hreflang="yue" data-title="子集公理" data-language-autonym="粵語" data-language-local-name="кантонски" class="interlanguage-link-target"><span>粵語</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-sr.svg" 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