CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="hu" dir="ltr"> <head> <meta charset="UTF-8"> <title>Naiv halmazelmélet – Wikipédia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )huwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"ymd","wgMonthNames":["","január","február","március","április","május","június","július","augusztus","szeptember","október","november","december"],"wgRequestId":"f5317eb0-34fd-4071-8497-0602aaab50ba","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Naiv_halmazelmélet","wgTitle":"Naiv halmazelmélet","wgCurRevisionId":26902028,"wgRevisionId":26902028,"wgArticleId":28129,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Halmazelmélet","Halmazelméleti axiómarendszerek és megalapozási paradigmák"],"wgPageViewLanguage":"hu","wgPageContentLanguage":"hu","wgPageContentModel":"wikitext","wgRelevantPageName":"Naiv_halmazelmélet","wgRelevantArticleId":28129,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true, "wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":2}}},"wgStableRevisionId":26902028,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"hu","pageLanguageDir":"ltr","pageVariantFallbacks":"hu"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":7000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q903783","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false,"wgSiteNoticeId":"2.195" };RLSTATE={"ext.gadget.infobox":"ready","ext.gadget.wikiMenuStyles":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready","ext.dismissableSiteNotice.styles":"ready"};RLPAGEMODULES=["site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.wdsearch","ext.gadget.irclogin","ext.gadget.ImageAnnotator.loader","ext.gadget.collapsible","ext.gadget.kepdia","ext.gadget.kinai","ext.gadget.poziciosTerkep", "ext.gadget.wikiMenu","ext.gadget.wiwosm","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","oojs-ui.styles.icons-media","oojs-ui-core.icons","wikibase.sidebar.tracking","ext.dismissableSiteNotice"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=hu&modules=ext.dismissableSiteNotice.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=hu&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=hu&modules=ext.gadget.infobox%2CwikiMenuStyles&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=hu&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Naiv halmazelmélet – Wikipédia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//hu.m.wikipedia.org/wiki/Naiv_halmazelm%C3%A9let"> <link rel="alternate" type="application/x-wiki" title="Szerkesztés" href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipédia (hu)"> <link rel="EditURI" type="application/rsd+xml" href="//hu.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://hu.wikipedia.org/wiki/Naiv_halmazelm%C3%A9let"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.hu"> <link rel="alternate" type="application/atom+xml" title="Wikipédia Atom-hírcsatorna" href="/w/index.php?title=Speci%C3%A1lis:Friss_v%C3%A1ltoztat%C3%A1sok&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Naiv_halmazelmélet rootpage-Naiv_halmazelmélet skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Ugrás a tartalomhoz</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Wiki"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Főmenü" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Főmenü</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Főmenü</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">elrejtés</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigáció </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Kezd%C5%91lap" title="A kezdőlap megtekintése [z]" accesskey="z"><span>Kezdőlap</span></a></li><li id="n-sidebar-contents" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Tartalom"><span>Tartalom</span></a></li><li id="n-sidebar-featured" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Kiemelt_sz%C3%B3cikkek_list%C3%A1ja"><span>Kiemelt szócikkek</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Friss_v%C3%A1ltoztat%C3%A1sok" title="A wikiben történt legutóbbi változtatások listája [r]" accesskey="r"><span>Friss változtatások</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Lap_tal%C3%A1lomra" title="Egy véletlenszerűen kiválasztott lap betöltése [x]" accesskey="x"><span>Lap találomra</span></a></li><li id="n-sidebar-enquiries" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Tudakoz%C3%B3"><span>Tudakozó</span></a></li> </ul> </div> </div> <div id="p-sidebar-participate" class="vector-menu mw-portlet mw-portlet-sidebar-participate" > <div class="vector-menu-heading"> Részvétel </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-sidebar-basics" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:%C3%9Aj_szerkeszt%C5%91knek"><span>Kezdőknek</span></a></li><li id="n-sidebar-help" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Seg%C3%ADts%C3%A9g"><span>Segítség</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Port%C3%A1l:K%C3%B6z%C3%B6ss%C3%A9g" title="A projektről, miben segíthetsz, mit hol találsz meg"><span>Közösségi portál</span></a></li><li id="n-sidebar-contact" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Kapcsolatfelv%C3%A9tel"><span>Kapcsolatfelvétel</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Kezd%C5%91lap" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipédia" src="/static/images/mobile/copyright/wikipedia-wordmark-fr.svg" style="width: 7.4375em; height: 1.125em;"> <img class="mw-logo-tagline" alt="" src="/static/images/mobile/copyright/wikipedia-tagline-hu.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Speci%C3%A1lis:Keres%C3%A9s" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Keresés a Wikipédián [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Keresés</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Keresés a Wikipédián" aria-label="Keresés a Wikipédián" autocapitalize="sentences" title="Keresés a Wikipédián [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Speciális:Keresés"> </div> <button class="cdx-button cdx-search-input__end-button">Keresés</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Személyes eszközök"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Megjelenés"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Megjelenés" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Megjelenés</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hu.wikipedia.org&uselang=hu" class=""><span>Adományok</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speci%C3%A1lis:Szerkeszt%C5%91i_fi%C3%B3k_l%C3%A9trehoz%C3%A1sa&returnto=Naiv+halmazelm%C3%A9let" title="Arra bíztatunk, hogy hozz létre egy fiókot, és jelentkezz be, azonban ez nem kötelező" class=""><span>Fiók létrehozása</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speci%C3%A1lis:Bel%C3%A9p%C3%A9s&returnto=Naiv+halmazelm%C3%A9let" title="Bejelentkezni javasolt, de nem kötelező [o]" accesskey="o" class=""><span>Bejelentkezés</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="További lehetőségek" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Személyes eszközök" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Személyes eszközök</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Felhasználói menü" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hu.wikipedia.org&uselang=hu"><span>Adományok</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Szerkeszt%C5%91i_fi%C3%B3k_l%C3%A9trehoz%C3%A1sa&returnto=Naiv+halmazelm%C3%A9let" title="Arra bíztatunk, hogy hozz létre egy fiókot, és jelentkezz be, azonban ez nem kötelező"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Fiók létrehozása</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Bel%C3%A9p%C3%A9s&returnto=Naiv+halmazelm%C3%A9let" title="Bejelentkezni javasolt, de nem kötelező [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Bejelentkezés</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Lapok kijelentkezett szerkesztőknek <a href="/wiki/Seg%C3%ADts%C3%A9g:Bevezet%C3%A9s" aria-label="Tudj meg többet a szerkesztésről"><span>további információk</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:K%C3%B6zrem%C5%B1k%C3%B6d%C3%A9seim" title="Erről az IP-címről végrehajtott szerkesztések listája [y]" accesskey="y"><span>Közreműködések</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Vit%C3%A1m" title="Az általad használt IP-címről végrehajtott szerkesztések megvitatása [n]" accesskey="n"><span>Vitalap</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eelrejt\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"hu\" dir=\"ltr\"\u003E\u003Cdiv style=\"padding:3px 10px; margin:10px; border:1px solid #b3b7ff; font-size:95%; text-align:center;\"\u003E\n\u003Cp\u003ENovember 30-án \u003Cb\u003E\u003Ca href=\"/wiki/Wikip%C3%A9dia:Wikital%C3%A1lkoz%C3%B3k/Budapest,_2024._november_30.\" title=\"Wikipédia:Wikitalálkozók/Budapest, 2024. november 30.\"\u003Etalálkozót tartunk\u003C/a\u003E\u003C/b\u003E Budapesten, ahol téged is szeretettel várunk!\n\u003C/p\u003E\n\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Wiki"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Tartalomjegyzék" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Tartalomjegyzék</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">elrejtés</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">Bevezető</div> </a> </li> <li id="toc-Története" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Története"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Története</span> </div> </a> <ul id="toc-Története-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_naiv_halmazelmélet_kiindulópontja" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_naiv_halmazelmélet_kiindulópontja"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>A naiv halmazelmélet kiindulópontja</span> </div> </a> <button aria-controls="toc-A_naiv_halmazelmélet_kiindulópontja-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>A(z) A naiv halmazelmélet kiindulópontja alszakasz kinyitása/becsukása</span> </button> <ul id="toc-A_naiv_halmazelmélet_kiindulópontja-sublist" class="vector-toc-list"> <li id="toc-Jelölés" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jelölés"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Jelölés</span> </div> </a> <ul id="toc-Jelölés-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Példa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Példa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Példa</span> </div> </a> <ul id="toc-Példa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ki_nem_mondott_feltételezések" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ki_nem_mondott_feltételezések"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ki nem mondott feltételezések</span> </div> </a> <ul id="toc-Ki_nem_mondott_feltételezések-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Az_ellentmondás" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Az_ellentmondás"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Az ellentmondás</span> </div> </a> <ul id="toc-Az_ellentmondás-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Források" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Források"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Források</span> </div> </a> <ul id="toc-Források-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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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">Naiv halmazelmélet</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="Ugrás egy más nyelvű szócikkre. Elérhető 31 nyelven" > <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-31" 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">31 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Naive_set_theory" title="Naive set theory – angol" lang="en" hreflang="en" data-title="Naive set theory" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%85%D8%AC%D9%85%D9%88%D8%B9%D8%A7%D8%AA_%D8%A7%D9%84%D9%85%D8%A8%D8%B3%D8%B7%D8%A9" title="نظرية المجموعات المبسطة – arab" lang="ar" hreflang="ar" data-title="نظرية المجموعات المبسطة" data-language-autonym="العربية" data-language-local-name="arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_informal_de_conxuntos" title="Teoría informal de conxuntos – asztúr" lang="ast" hreflang="ast" data-title="Teoría informal de conxuntos" data-language-autonym="Asturianu" data-language-local-name="asztúr" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_informal_de_conjunts" title="Teoria informal de conjunts – katalán" lang="ca" hreflang="ca" data-title="Teoria informal de conjunts" data-language-autonym="Català" data-language-local-name="katalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Naivn%C3%AD_teorie_mno%C5%BEin" title="Naivní teorie množin – cseh" lang="cs" hreflang="cs" data-title="Naivní teorie množin" data-language-autonym="Čeština" data-language-local-name="cseh" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Naive_Mengenlehre" title="Naive Mengenlehre – német" lang="de" hreflang="de" data-title="Naive Mengenlehre" data-language-autonym="Deutsch" data-language-local-name="német" 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%91%CF%86%CE%B5%CE%BB%CE%AE%CF%82_%CF%83%CF%85%CE%BD%CE%BF%CE%BB%CE%BF%CE%B8%CE%B5%CF%89%CF%81%CE%AF%CE%B1" title="Αφελής συνολοθεωρία – görög" lang="el" hreflang="el" data-title="Αφελής συνολοθεωρία" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Naiva_aroteorio" title="Naiva aroteorio – eszperantó" lang="eo" hreflang="eo" data-title="Naiva aroteorio" data-language-autonym="Esperanto" data-language-local-name="eszperantó" 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/Teor%C3%ADa_informal_de_conjuntos" title="Teoría informal de conjuntos – spanyol" lang="es" hreflang="es" data-title="Teoría informal de conjuntos" data-language-autonym="Español" data-language-local-name="spanyol" 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/Naiivne_hulgateooria" title="Naiivne hulgateooria – észt" lang="et" hreflang="et" data-title="Naiivne hulgateooria" data-language-autonym="Eesti" data-language-local-name="észt" 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/Multzoen_teoria_informal" title="Multzoen teoria informal – baszk" lang="eu" hreflang="eu" data-title="Multzoen teoria informal" data-language-autonym="Euskara" data-language-local-name="baszk" 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/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C_%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87%E2%80%8C%D9%87%D8%A7" title="نظریه طبیعی مجموعه‌ها – perzsa" lang="fa" hreflang="fa" data-title="نظریه طبیعی مجموعه‌ها" data-language-autonym="فارسی" data-language-local-name="perzsa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_na%C3%AFve_des_ensembles" title="Théorie naïve des ensembles – francia" lang="fr" hreflang="fr" data-title="Théorie naïve des ensembles" data-language-autonym="Français" data-language-local-name="francia" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_informal_de_conxuntos" title="Teoría informal de conxuntos – gallego" lang="gl" hreflang="gl" data-title="Teoría informal de conxuntos" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%91%D7%95%D7%A6%D7%95%D7%AA_%D7%94%D7%A0%D7%90%D7%99%D7%91%D7%99%D7%AA" title="תורת הקבוצות הנאיבית – héber" lang="he" hreflang="he" data-title="תורת הקבוצות הנאיבית" data-language-autonym="עברית" data-language-local-name="héber" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Naivna_teorija_skupova" title="Naivna teorija skupova – horvát" lang="hr" hreflang="hr" data-title="Naivna teorija skupova" data-language-autonym="Hrvatski" data-language-local-name="horvát" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_ingenua_degli_insiemi" title="Teoria ingenua degli insiemi – olasz" lang="it" hreflang="it" data-title="Teoria ingenua degli insiemi" data-language-autonym="Italiano" data-language-local-name="olasz" 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%B4%A0%E6%9C%B4%E9%9B%86%E5%90%88%E8%AB%96" title="素朴集合論 – japán" lang="ja" hreflang="ja" data-title="素朴集合論" data-language-autonym="日本語" data-language-local-name="japán" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%86%8C%EB%B0%95%ED%95%9C_%EC%A7%91%ED%95%A9%EB%A1%A0" title="소박한 집합론 – koreai" lang="ko" hreflang="ko" data-title="소박한 집합론" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B8%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D1%82%D0%B0" title="Наивна теорија на множествата – macedón" lang="mk" hreflang="mk" data-title="Наивна теорија на множествата" data-language-autonym="Македонски" data-language-local-name="macedón" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9B%E1%80%AD%E1%80%AF%E1%80%B8%E1%80%9B%E1%80%AD%E1%80%AF%E1%80%B8%E1%80%A1%E1%80%85%E1%80%AF%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE" title="ရိုးရိုးအစုသီအိုရီ – burmai" lang="my" hreflang="my" data-title="ရိုးရိုးအစုသီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmai" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Na%C3%AFeve_verzamelingenleer" title="Naïeve verzamelingenleer – holland" lang="nl" hreflang="nl" data-title="Naïeve verzamelingenleer" data-language-autonym="Nederlands" data-language-local-name="holland" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_ing%C3%AAnua_dos_conjuntos" title="Teoria ingênua dos conjuntos – portugál" lang="pt" hreflang="pt" data-title="Teoria ingênua dos conjuntos" data-language-autonym="Português" data-language-local-name="portugál" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B8%D0%B2%D0%BD%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="Наивная теория множеств – orosz" lang="ru" hreflang="ru" data-title="Наивная теория множеств" data-language-autonym="Русский" data-language-local-name="orosz" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Naive_set_theory" title="Naive set theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Naive set theory" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Naivna_teorija_skupova" title="Naivna teorija skupova – szerb" lang="sr" hreflang="sr" data-title="Naivna teorija skupova" data-language-autonym="Српски / srpski" data-language-local-name="szerb" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Naif_k%C3%BCme_teorisi" title="Naif küme teorisi – török" lang="tr" hreflang="tr" data-title="Naif küme teorisi" data-language-autonym="Türkçe" data-language-local-name="török" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%97%D0%B2%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD" title="Наївна теорія множин – ukrán" lang="uk" hreflang="uk" data-title="Наївна теорія множин" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_t%E1%BA%ADp_h%E1%BB%A3p_ng%C3%A2y_th%C6%A1" title="Lý thuyết tập hợp ngây thơ – vietnámi" lang="vi" hreflang="vi" data-title="Lý thuyết tập hợp ngây thơ" data-language-autonym="Tiếng Việt" data-language-local-name="vietnámi" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9C%B4%E7%B4%A0%E9%9B%86%E5%90%88%E8%AE%BA" title="朴素集合论 – kínai" lang="zh" hreflang="zh" data-title="朴素集合论" data-language-autonym="中文" data-language-local-name="kínai" 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/%E6%A8%B8%E7%B4%A0%E9%9B%86%E5%90%88%E8%AB%96" title="樸素集合論 – kantoni" lang="yue" hreflang="yue" data-title="樸素集合論" data-language-autonym="粵語" data-language-local-name="kantoni" 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/Q903783#sitelinks-wikipedia" title="Nyelvközi hivatkozások szerkesztése" class="wbc-editpage">Hivatkozások szerkesztése</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="Névterek"> <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/Naiv_halmazelm%C3%A9let" title="A lap megtekintése [c]" accesskey="c"><span>Szócikk</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Vita:Naiv_halmazelm%C3%A9let" rel="discussion" title="Az oldal tartalmának megvitatása [t]" accesskey="t"><span>Vitalap</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="Nyelvvariáns váltása" > <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">magyar</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="Nézetek"> <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/Naiv_halmazelm%C3%A9let"><span>Olvasás</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=history" title="A lap korábbi változatai [h]" accesskey="h"><span>Laptörténet</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <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="Eszközök" > <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">Eszközök</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">Eszközök</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">elrejtés</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="További lehetőségek" > <div class="vector-menu-heading"> Műveletek </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/Naiv_halmazelm%C3%A9let"><span>Olvasás</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=history"><span>Laptörténet</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Általános </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Mi_hivatkozik_erre/Naiv_halmazelm%C3%A9let" title="Az erre a lapra hivatkozó más lapok listája [j]" accesskey="j"><span>Mi hivatkozik erre?</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Kapcsol%C3%B3d%C3%B3_v%C3%A1ltoztat%C3%A1sok/Naiv_halmazelm%C3%A9let" rel="nofollow" title="Az erről a lapról hivatkozott lapok utolsó változtatásai [k]" accesskey="k"><span>Kapcsolódó változtatások</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Speci%C3%A1lis_lapok" title="Az összes speciális lap listája [q]" accesskey="q"><span>Speciális lapok</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&oldid=26902028" title="Állandó hivatkozás ezen lap ezen változatához"><span>Hivatkozás erre a változatra</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=info" title="További információk erről a lapról"><span>Lapinformációk</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Hivatkoz%C3%A1s&page=Naiv_halmazelm%C3%A9let&id=26902028&wpFormIdentifier=titleform" title="Információk a lap idézésével kapcsolatban"><span>Hogyan hivatkozz erre a lapra?</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:UrlShortener&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FNaiv_halmazelm%25C3%25A9let"><span>Rövidített URL készítése</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:QrCode&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FNaiv_halmazelm%25C3%25A9let"><span>QR-kód letöltése</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"> Nyomtatás/exportálás </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=Speci%C3%A1lis:K%C3%B6nyv&bookcmd=book_creator&referer=Naiv+halmazelm%C3%A9let"><span>Könyv készítése</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:DownloadAsPdf&page=Naiv_halmazelm%C3%A9let&action=show-download-screen"><span>Letöltés PDF-ként</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&printable=yes" title="A lap nyomtatható változata [p]" accesskey="p"><span>Nyomtatható változat</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"> Társprojektek </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q903783" title="Kapcsolt adattárelem [g]" accesskey="g"><span>Wikidata-adatlap</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="Oldal eszközök"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Megjelenés"> <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">Megjelenés</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">elrejtés</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-indicator-fr-review-status" class="mw-indicator"><indicator name="fr-review-status" class="mw-fr-review-status-indicator" id="mw-fr-revision-toggle"><span class="cdx-fr-css-icon-review--status--stable"></span><b>Ellenőrzött</b></indicator></div> </div> <div id="siteSub" class="noprint">A Wikipédiából, a szabad enciklopédiából</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div id="mw-fr-revision-messages"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">Változat állapota</h2><p class="cdx-dialog__header__subtitle">Ez a lap egy ellenőrzött változata</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Ez a <a href="/wiki/Wikip%C3%A9dia:Jel%C3%B6lt_lapv%C3%A1ltozatok" title="Wikipédia:Jelölt lapváltozatok">közzétett változat</a>, <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Rendszernapl%C3%B3k&type=review&page=Naiv_halmazelm%C3%A9let">ellenőrizve</a>: <i>2024. február 20.</i><p><table id="mw-fr-revisionratings-box" class="flaggedrevs-color-1" style="margin: auto;" cellpadding="0"><tr><td class="fr-text" style="vertical-align: middle;">Pontosság</td><td class="fr-value40" style="vertical-align: middle;">ellenőrzött</td></tr></table></p></div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hu" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r27515026">.mw-parser-output .plainlist ul{line-height:inherit;list-style:none none;margin:0;padding:0}.mw-parser-output .plainlist ul li{margin-bottom:0}</style><table class="navbox" style="width: 280px; margin: 0px 0px 10px 10px; float: right; clear:right;"><tbody><tr><th style="background-color:#ccccff;"><a href="/wiki/Matematika" title="Matematika">Matematika</a></th></tr><tr><td style="text-align:center; background-color:#ccccff;"><b>A matematika alapjai</b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Halmazelm%C3%A9let" title="Halmazelmélet">Halmazelmélet</a></li> <li><a class="mw-selflink selflink">Naiv halmazelmélet</a></li> <li><a href="/wiki/Axiomatikus_halmazelm%C3%A9let" title="Axiomatikus halmazelmélet">Axiomatikus halmazelmélet</a></li> <li><a href="/wiki/Matematikai_logika" title="Matematikai logika">Matematikai logika</a></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><b><a href="/wiki/Algebra" title="Algebra">Algebra</a></b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Elemi_algebra" title="Elemi algebra">Elemi algebra</a></li> <li><a href="/wiki/Line%C3%A1ris_algebra" title="Lineáris algebra">Lineáris algebra</a></li> <li><a href="/wiki/Polinom" title="Polinom">Polinomok</a></li> <li><a href="/wiki/Absztrakt_algebra" title="Absztrakt algebra">Absztrakt algebra</a></li> <li><a href="/wiki/Csoportelm%C3%A9let" title="Csoportelmélet">Csoportelmélet</a></li> <li><a href="/wiki/Gy%C5%B1r%C5%B1_(matematika)" title="Gyűrű (matematika)">Gyűrűelmélet</a></li> <li><a href="/wiki/Testelm%C3%A9let" title="Testelmélet">Testelmélet</a></li> <li><a href="/wiki/M%C3%A1trix_(matematika)" title="Mátrix (matematika)">Mátrixok</a></li> <li><a href="/w/index.php?title=Univerz%C3%A1lis_algebra&action=edit&redlink=1" class="new" title="Univerzális algebra (a lap nem létezik)">Univerzális algebra</a></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><b><a href="/wiki/Matematikai_anal%C3%ADzis" title="Matematikai analízis">Analízis</a></b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Val%C3%B3s_anal%C3%ADzis" title="Valós analízis">Valós analízis</a></li> <li><a href="/wiki/Komplex_anal%C3%ADzis" title="Komplex analízis">Komplex analízis</a></li> <li><a href="/wiki/Vektoranal%C3%ADzis" title="Vektoranalízis">Vektoranalízis</a></li> <li><a href="/wiki/Differenci%C3%A1legyenlet" title="Differenciálegyenlet">Differenciálegyenletek</a></li> <li><a href="/wiki/Funkcion%C3%A1lanal%C3%ADzis" title="Funkcionálanalízis">Funkcionálanalízis</a></li> <li><a href="/wiki/M%C3%A9rt%C3%A9k_(matematika)" title="Mérték (matematika)">Mértékelmélet</a></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><b><a href="/wiki/Geometria" title="Geometria">Geometria</a></b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Euklideszi_geometria" title="Euklideszi geometria">Euklideszi geometria</a></li> <li><a href="/wiki/Nemeuklideszi_geometria" title="Nemeuklideszi geometria">Nemeuklideszi geometria</a></li> <li><a href="/wiki/Affin_geometria" title="Affin geometria">Affin geometria</a></li> <li><a href="/wiki/Projekt%C3%ADv_geometria" title="Projektív geometria">Projektív geometria</a></li> <li><a href="/wiki/Differenci%C3%A1lgeometria" title="Differenciálgeometria">Differenciálgeometria</a></li> <li><a href="/w/index.php?title=Algebrai_geometria&action=edit&redlink=1" class="new" title="Algebrai geometria (a lap nem létezik)">Algebrai geometria</a></li> <li><a href="/wiki/Topol%C3%B3gia" title="Topológia">Topológia</a></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><b><a href="/wiki/Sz%C3%A1melm%C3%A9let" title="Számelmélet">Számelmélet</a></b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Algebrai_sz%C3%A1melm%C3%A9let" title="Algebrai számelmélet">Algebrai számelmélet</a></li> <li><a href="/wiki/Analitikus_sz%C3%A1melm%C3%A9let" class="mw-redirect" title="Analitikus számelmélet">Analitikus számelmélet</a></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><b><a href="/wiki/Diszkr%C3%A9t_matematika" title="Diszkrét matematika">Diszkrét matematika</a></b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Kombinatorika" title="Kombinatorika">Kombinatorika</a></li> <li><a href="/wiki/Gr%C3%A1felm%C3%A9let" title="Gráfelmélet">Gráfelmélet</a></li> <li><a href="/wiki/J%C3%A1t%C3%A9kelm%C3%A9let" title="Játékelmélet">Játékelmélet</a></li> <li><a href="/wiki/Algoritmus" title="Algoritmus">Algoritmusok</a></li> <li><a href="/wiki/Form%C3%A1lis_nyelv" title="Formális nyelv">Formális nyelvek</a></li> <li><a href="/wiki/Inform%C3%A1ci%C3%B3elm%C3%A9let" title="Információelmélet">Információelmélet</a></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><b><a href="/w/index.php?title=Alkalmazott_matematika&action=edit&redlink=1" class="new" title="Alkalmazott matematika (a lap nem létezik)">Alkalmazott matematika</a></b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Numerikus_anal%C3%ADzis" title="Numerikus analízis">Numerikus analízis</a></li> <li><a href="/wiki/Val%C3%B3sz%C3%ADn%C5%B1s%C3%A9gsz%C3%A1m%C3%ADt%C3%A1s" title="Valószínűségszámítás">Valószínűségszámítás</a></li> <li><a href="/wiki/Statisztika" title="Statisztika">Statisztika</a></li> <li><a href="/wiki/K%C3%A1oszelm%C3%A9let" title="Káoszelmélet">Káoszelmélet</a></li> <li><a href="/w/index.php?title=Matematikai_fizika&action=edit&redlink=1" class="new" title="Matematikai fizika (a lap nem létezik)">Matematikai fizika</a></li> <li><a href="/wiki/Biomatematika" title="Biomatematika">Matematikai biológia</a></li> <li><a href="/wiki/Matematikai_k%C3%B6zgazdas%C3%A1gtan" title="Matematikai közgazdaságtan">Gazdasági matematika</a></li> <li><a href="/wiki/Kriptogr%C3%A1fia" title="Kriptográfia">Kriptográfia</a></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><b>Általános</b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Matematikus" title="Matematikus">Matematikusok</a></li> <li><a href="/wiki/A_matematika_t%C3%B6rt%C3%A9nete" title="A matematika története">Matematikatörténet</a></li> <li><a href="/wiki/Matematikafiloz%C3%B3fia" title="Matematikafilozófia">Matematikafilozófia</a></li> <li><small><a href="/wiki/Port%C3%A1l:Matematika" title="Portál:Matematika">Portál</a></small></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><div class="navbar noprint hlist plainlinks mini" style="display:inline;font-size:xx-small"><style data-mw-deduplicate="TemplateStyles:r26593303">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><span style="display:none"><a href="/wiki/Sablon:Matematika" title="Sablon:Matematika">Sablon:Matematika</a></span><ul style="display:inline"><li class="nv-view"><a class="external text" href="https://hu.wikipedia.org/wiki/Sablon:Matematika"><span title="Mutasd ezt a sablont">m</span></a></li> <li class="nv-talk"><a class="external text" href="https://hu.wikipedia.org/wiki/Sablonvita:Matematika"><span title="A sablon vitalapja">v</span></a></li> <li class="nv-edit"><a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Sablon:Matematika&action=edit"><span title="A sablon szerkesztése">sz</span></a></li></ul></div></td></tr></tbody></table> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Története"><span id="T.C3.B6rt.C3.A9nete"></span>Története</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit&section=1" title="Szakasz szerkesztése: Története"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Halmazelm%C3%A9let" title="Halmazelmélet">halmazelmélet</a> alapjait <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> rakta le egy 1874-ben megjelent cikkében, melyben a <a href="/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok">valós számok</a> nem <a href="/wiki/Megsz%C3%A1ml%C3%A1lhat%C3%B3an_v%C3%A9gtelen" class="mw-redirect" title="Megszámlálhatóan végtelen">megszámlálhatóan végtelen</a> voltát bizonyította be elsőként. Cantor gondolata az volt, hogy ne csak számok, pontok, egyenesek összességeit tekintsük, hanem ezek összességeinek összességeit, … is. Ekkor összességek végtelen hierarchiáját alkotjuk meg gondolatban, ami érdekes matematikai és filozófiai problémákat vet fel. Az 1874-es cikk eredménye azért megdöbbentő, mert kiderül: ugyan természetes számból és valós számból is végtelen sok van, de mégis valamilyen szempontból a valós számok összessége „magasabbrendűen” (nem megszámlálható módon) végtelen, mint ahogy a természetes számok összessége végtelen, sőt, ahogy számból, úgy végtelenből is végtelen sok van. Cantor ezzel megteremtette a végtelen <a href="/wiki/Sz%C3%A1moss%C3%A1g" title="Számosság">számosságok</a> elméletét. Az összességre a <i>Menge</i> német szót használta, később más elnevezések is napvilágot láttak; a magyar nyelvben a <a href="/wiki/Halmaz_(matematika)" title="Halmaz (matematika)">halmaz</a> szót használják matematikai szakkifejezésként. </p><p>Eredményeit <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a>, <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a> és <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> is felhasználta. Szerencsétlenségükre Russell munkája során felfedezett egy ellentmondást, mely Cantor alapgondolatából következik (ez a <a href="/wiki/Russell-paradoxon" title="Russell-paradoxon">Russell-paradoxon</a>) és azt levélben meg is küldte Fregenek, aki ezt az érvelést az éppen nyomdába készülő könyvének utószavába be is illesztette. Ezzel 1903-ban napvilágot látott Cantor halmazelméletének ellentmondásossága. Azóta nevezik Cantor elméletét naiv (azaz kezdetleges) halmazelméletnek. (Valójában Cantor – ahogy rajta kívül sokan mások is – felfedezett egy ellentmondást, ezt <a href="/wiki/Cantor-paradoxon" title="Cantor-paradoxon">Cantor-paradoxon</a> néven emlegetik.) A halmazelméletet sikerült az axiomatikus módszer segítségével megmenteni és az ismert ellentmondásaitól megszabadítani. A korban a feladatot Russell (a <a href="/w/index.php?title=T%C3%ADpuselm%C3%A9let&action=edit&redlink=1" class="new" title="Típuselmélet (a lap nem létezik)">típuselméletben</a>), <a href="/w/index.php?title=Zermelo&action=edit&redlink=1" class="new" title="Zermelo (a lap nem létezik)">Zermelo</a> és <a href="/w/index.php?title=Fraenkel&action=edit&redlink=1" class="new" title="Fraenkel (a lap nem létezik)">Fraenkel</a> (a <a href="/w/index.php?title=Zermelo%E2%80%93Fraenkel-halmazelm%C3%A9let&action=edit&redlink=1" class="new" title="Zermelo–Fraenkel-halmazelmélet (a lap nem létezik)">Zermelo–Fraenkel-halmazelméletben</a>) és az <a href="/wiki/Intuicionizmus" class="mw-redirect" title="Intuicionizmus">intuicionisták</a> a fajták elméletében oldották meg. Később más axiomatikus halmazelméletek is születtek (például a <a href="/wiki/Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del-halmazelm%C3%A9let" title="Neumann–Bernays–Gödel-halmazelmélet">Neumann–Bernays–Gödel-halmazelmélet</a> és a <a href="/wiki/Bourbaki-csoport" title="Bourbaki-csoport">Bourbaki</a>-halmazelmélet). </p> <div class="mw-heading mw-heading2"><h2 id="A_naiv_halmazelmélet_kiindulópontja"><span id="A_naiv_halmazelm.C3.A9let_kiindul.C3.B3pontja"></span>A naiv halmazelmélet kiindulópontja</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit&section=2" title="Szakasz szerkesztése: A naiv halmazelmélet kiindulópontja"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A naiv halmazelmélet hallgatólagos alapfeltevése volt, hogy ha <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ffc98a039ed280ea6420a0f53758bddef8d9a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle {T}}"></span> valamilyen tulajdonság, akkor gondolhatunk mindazon dolgok összességére, melyekre a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ffc98a039ed280ea6420a0f53758bddef8d9a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle {T}}"></span> tulajdonság teljesül. Ezt az összességet a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ffc98a039ed280ea6420a0f53758bddef8d9a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle {T}}"></span> tulajdonság <i>igazságtartományának</i> nevezzük. </p> <div class="mw-heading mw-heading3"><h3 id="Jelölés"><span id="Jel.C3.B6l.C3.A9s"></span>Jelölés</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit&section=3" title="Szakasz szerkesztése: Jelölés"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Magát a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ffc98a039ed280ea6420a0f53758bddef8d9a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle {T}}"></span> tulajdonságot gyakran funkcionális jelölésmódban úgy jelöljük, hogy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a192e2a47974581ac039c50c0024627a0d10ec81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle {T(x)}}"></span>. Itt az <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span> karaktert <i>változó</i>nak nevezzük és azt jelképezi, hogy a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a192e2a47974581ac039c50c0024627a0d10ec81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle {T(x)}}"></span> kifejezés <i>nyitott mondat</i>, igazságértéke még nem értelmezhető. <i>Zárt</i> kijelentő mondat – azaz olyan, melynek létezik igaz vagy hamis értéke – csak akkor lesz belőle, ha az <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span> változó helyére valamilyen dolog nevét helyettesítjük. </p><p>A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a192e2a47974581ac039c50c0024627a0d10ec81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle {T(x)}}"></span> tulajdonság igazságtartományát </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 \{x\mid T(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\mid T(x)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2734966a3c5ae6d8cafea2a256d22aa6cde96a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.367ex; height:2.843ex;" alt="{\displaystyle \{x\mid T(x)\}}"></span></dd></dl> <p>-szel jelöljük és úgy mondjuk ki, hogy „azon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span>-ek összessége, melyre a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a192e2a47974581ac039c50c0024627a0d10ec81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle {T(x)}}"></span> tulajdonság igaz”. </p> <div class="mw-heading mw-heading3"><h3 id="Példa"><span id="P.C3.A9lda"></span>Példa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit&section=4" title="Szakasz szerkesztése: Példa"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Legyen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ffc98a039ed280ea6420a0f53758bddef8d9a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle {T}}"></span> : „kutya” . Funkcionális jelölésmódban <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(x):}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(x):}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7427ceadba7605e8d3cc67ae6f1a4ee470c5959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.067ex; height:2.843ex;" alt="{\displaystyle {T(x):}}"></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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span> kutya”. Ekkor „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span> kutya” még nyitott mondat, zártat úgy képezhetünk belőle, ha az <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span> változó helyére például <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {Buksi}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>u</mi> <mi>k</mi> <mi>s</mi> <mi>i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Buksi}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640e2052321353c1847f552ae68ab71346755c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.198ex; height:2.176ex;" alt="{\displaystyle {Buksi}}"></span>, a kutya vagy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {Cirmi}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>i</mi> <mi>r</mi> <mi>m</mi> <mi>i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Cirmi}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c15a6ff4fa3f1083f899325fe7d033ce3617f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.46ex; height:2.176ex;" alt="{\displaystyle {Cirmi}}"></span>, a macska nevét helyettesítjük. Ekkor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(Buksi)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mi>u</mi> <mi>k</mi> <mi>s</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(Buksi)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d286d8f838926a135aa2bf1c1e6a388606067c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.643ex; height:2.843ex;" alt="{\displaystyle {T(Buksi)}}"></span> egy, a valóságnak megfelelő állapotot leíró, tehát igaz mondat, míg <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(Cirmi)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>i</mi> <mi>r</mi> <mi>m</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(Cirmi)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59ec1a83e69419341b36a3bc8fc947768897ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.906ex; height:2.843ex;" alt="{\displaystyle {T(Cirmi)}}"></span> nem felel meg a valóságnak, így hamis. Végeredményben képezhetjük a kutyák összességét: </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 \{x\mid T(x)\}=\{x\mid x{\mbox{ kutya}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> kutya</mtext> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\mid T(x)\}=\{x\mid x{\mbox{ kutya}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adaf7e4a5e4d164b57735d7ed9294ad7be1b4c85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.782ex; height:2.843ex;" alt="{\displaystyle \{x\mid T(x)\}=\{x\mid x{\mbox{ kutya}}\}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ki_nem_mondott_feltételezések"><span id="Ki_nem_mondott_felt.C3.A9telez.C3.A9sek"></span>Ki nem mondott feltételezések</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit&section=5" title="Szakasz szerkesztése: Ki nem mondott feltételezések"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eddigi fejtegetésünk a <a href="/wiki/Logikai_grammatika" title="Logikai grammatika">logikai grammatika</a> témakörébe tartozik és legfeljebb az „igaznak lenni” minősítés homályos értelmezése felől támadható. Ma már tudjuk, hogy Cantor a fentieken felül kimondatlanul feltételezte a következőket: </p> <ol><li><b>A komprehenzivitás elve:</b> akármilyen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a192e2a47974581ac039c50c0024627a0d10ec81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle {T(x)}}"></span> tulajdonság esetén, az <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span> változó helyére minden dolog nevét írhatjuk, és összegyűjthetjük az { <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> | <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1171c29b4c2b5575f50a4ea9313f90448a2cbe05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}"></span> } szimbólum alá az összes olyan dolgot mely teljesíti a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {T(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a192e2a47974581ac039c50c0024627a0d10ec81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle {T(x)}}"></span> tulajdonságot.</li> <li><b>Az extenzionalitás elve:</b> Két összesség akkor és csak akkor egyenlő, ha elemeik megegyeznek.</li></ol> <p>Cantor a <i>Menge</i>, azaz halmaz szót használta a { <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> | <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1171c29b4c2b5575f50a4ea9313f90448a2cbe05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}"></span> } összesség megnevezésére. Ha valamely <span class="mwe-math-element"><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"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </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/ecaee8e0957dc3d0a7a3c7da3b54def4bcd27062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle {a}}"></span> dolog benne van a { <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> | <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1171c29b4c2b5575f50a4ea9313f90448a2cbe05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}"></span> } halmazban, akkor ezt szimbolikusan így jelöljük: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ∈ { <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> | <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1171c29b4c2b5575f50a4ea9313f90448a2cbe05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}"></span> }. </p> <div class="mw-heading mw-heading2"><h2 id="Az_ellentmondás"><span id="Az_ellentmond.C3.A1s"></span>Az ellentmondás</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit&section=6" title="Szakasz szerkesztése: Az ellentmondás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Russell-paradoxon" title="Russell-paradoxon">Russell-paradoxon</a> feloldását mások máshogy képzelték. <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a> abban látta az ellentmondás fellépésének okát, hogy az összességekre – úgy tűnik – nem áll a kizárt harmadik elve. Russell maga szükségesnek tartotta szigorúan megkülönböztetni a dolgokat, a dolgok összességeitől. A Russell-paradoxon mindazonáltal a következők miatt lép fel. Ellentmondások hátterében gyakran az önmaguk igazságára hivatkozó mondatok állnak. Ez húzódik meg <a href="/wiki/A_hazug_paradoxona" title="A hazug paradoxona">a hazug paradoxona</a> mögött, a <a href="/w/index.php?title=G%C3%B6del-f%C3%A9le_nemteljess%C3%A9gi_t%C3%A9tele&action=edit&redlink=1" class="new" title="Gödel-féle nemteljességi tétele (a lap nem létezik)">Gödel-féle nemteljességi tételekben</a> és ez ad alapot a hatványhalmaz számosságára vonatkozó tétel (a Cantor-tétel) fennállására. Mivel az <span class="mwe-math-element"><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\notin x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∉</mo> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x\notin x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1319cb1fc45b88fd760ddd526e19410aff57eac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.5ex; height:2.676ex;" alt="{\displaystyle {x\notin x}}"></span> kijelentésben összességek is szerepelhetnek és az összességeket egyértelműen meghatározza a definiáló tulajdonságuk, így a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x\notin x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∉</mo> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x\notin x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1319cb1fc45b88fd760ddd526e19410aff57eac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.5ex; height:2.676ex;" alt="{\displaystyle {x\notin x}}"></span> kijelentésből könnyen csinálhatunk saját magára hivatkozó mondatot: </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 R=\{x\mid x\notin x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mo>∉</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\{x\mid x\notin x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ee4558d62066d3d8eb7b2164830a30a88ecc2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.954ex; height:2.843ex;" alt="{\displaystyle R=\{x\mid x\notin x\}}"></span> azaz</dd></dl> <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\in R\Leftrightarrow x\notin x}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>R</mi> <mo stretchy="false">⇔</mo> <mi>x</mi> <mo>∉</mo> <mi>x</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x\in R\Leftrightarrow x\notin x}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ffbeea21de3e1dcad0498716a8b80c963c7c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:15.436ex; height:2.676ex;" alt="{\displaystyle {x\in R\Leftrightarrow x\notin x}\,\!}"></span>, így <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span>-ben saját magát <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3abe7f424f59ed9925c7f622af7aead87831412b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle {R}}"></span>-et szerepeltetve:</dd></dl> <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 {R\in R\Leftrightarrow R\notin R}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo>∈</mo> <mi>R</mi> <mo stretchy="false">⇔</mo> <mi>R</mi> <mo>∉</mo> <mi>R</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {R\in R\Leftrightarrow R\notin R}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f46826feccb23de5f7db146940617ada9ed0c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:16.739ex; height:2.676ex;" alt="{\displaystyle {R\in R\Leftrightarrow R\notin R}\,\!}"></span></dd></dl> <p>Ez utóbbi módszert, amikor egy tulajdonság változójának helyébe magát a tulajdonságot (pontosabban annak megnevezését) helyettesítjük, <i>Cantor-féle átlós eljárásnak</i> nevezzük. A sors fintora, hogy Cantor halmazelméletén pont a saját maga által először alkalmazott eljárás segítségével tudott Russell rést ütni. </p> <div class="mw-heading mw-heading2"><h2 id="Források"><span id="Forr.C3.A1sok"></span>Források</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naiv_halmazelm%C3%A9let&action=edit&section=7" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Robert Goldblatt, <i>TOPOI – The categorical analysis of logic</i>, North-Holland Publ. Co., 1984 <a rel="nofollow" class="external text" href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&view=75&frames=0&seq=3">elektronikus könyvtári formában itt</a></li> <li><a href="/wiki/Ruzsa_Imre" title="Ruzsa Imre">Ruzsa Imre</a> – Máté András, <i>Bevezetés a modern logikába</i>, Osiris Kiadó, 1997.</li> <li>Gottlob Frege, <i>Az aritmetika alaptörvényei II.</i>, Utószó (1903), in: <i>Gottlob Frege, Logikai vizsgálódások – Válogatott tanulmányok</i>, szerk.: Máté András, Osiris Kiadó, 2000.</li></ul> <div class="noprint noviewer" style="overflow: hidden; clear: both;"><div style="margin-left:0; margin-right:2px;"><ul style="display:block; list-style-image:none; list-style-type:none; width:100%; vertical-align:middle; margin:0; padding:0; min-height: 27px;"><li style="float:left; min-height: 27px; line-height:25px; width:100%; margin:0; margin-top:.5em; margin-left:0; margin-right:0; padding:0; border:1px solid #CCF; background-color:#F0EEFF"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:P_cartesian_graph.svg" class="mw-file-description" title="matematika"><img alt="matematika" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/25px-P_cartesian_graph.svg.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/38px-P_cartesian_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/50px-P_cartesian_graph.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> <b><a href="/wiki/Port%C3%A1l:Matematika" title="Portál:Matematika">Matematikaportál</a></b> • összefoglaló, színes tartalomajánló lap</li></ul></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">A lap eredeti címe: „<a dir="ltr" href="https://hu.wikipedia.org/w/index.php?title=Naiv_halmazelmélet&oldid=26902028">https://hu.wikipedia.org/w/index.php?title=Naiv_halmazelmélet&oldid=26902028</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikip%C3%A9dia:Kateg%C3%B3ri%C3%A1k" title="Wikipédia:Kategóriák">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Halmazelm%C3%A9let" title="Kategória:Halmazelmélet">Halmazelmélet</a></li><li><a href="/wiki/Kateg%C3%B3ria:Halmazelm%C3%A9leti_axi%C3%B3marendszerek_%C3%A9s_megalapoz%C3%A1si_paradigm%C3%A1k" title="Kategória:Halmazelméleti axiómarendszerek és megalapozási paradigmák">Halmazelméleti axiómarendszerek és megalapozási paradigmák</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> A lap utolsó módosítása: 2024. február 20., 11:47</li> <li id="footer-info-copyright">A lap szövege <a rel="nofollow" class="external text" href="http://creativecommons.org/licenses/by-sa/4.0/deed.hu">Creative Commons Nevezd meg! – Így add tovább! 4.0</a> licenc alatt van; egyes esetekben más módon is felhasználható. Részletekért lásd a <a href="/wiki/Wikip%C3%A9dia:Felhaszn%C3%A1l%C3%A1si_felt%C3%A9telek" title="Wikipédia:Felhasználási feltételek">felhasználási feltételeket</a>.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Adatvédelmi irányelvek</a></li> <li id="footer-places-about"><a href="/wiki/Wikip%C3%A9dia:R%C3%B3lunk">A Wikipédiáról</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikip%C3%A9dia:Jogi_nyilatkozat">Jogi nyilatkozat</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Magatartási kódex</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Fejlesztők</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/hu.wikipedia.org">Statisztikák</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Sütinyilatkozat</a></li> <li id="footer-places-mobileview"><a href="//hu.m.wikipedia.org/w/index.php?title=Naiv_halmazelm%C3%A9let&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobil nézet</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-7fc47fc68d-qrqh9","wgBackendResponseTime":506,"wgPageParseReport":{"limitreport":{"cputime":"0.125","walltime":"0.234","ppvisitednodes":{"value":619,"limit":1000000},"postexpandincludesize":{"value":13193,"limit":2097152},"templateargumentsize":{"value":3188,"limit":2097152},"expansiondepth":{"value":10,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":4201,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 89.213 1 -total"," 83.21% 74.236 1 Sablon:Matematika"," 78.82% 70.321 1 Sablon:Navoszlop"," 66.98% 59.759 1 Sablon:M-v-sz"," 15.58% 13.899 1 Sablon:Portál"," 3.37% 3.008 2 Sablon:Portál/speciális"," 3.11% 2.776 1 Portál:Matematika/ikon"," 2.75% 2.451 1 Sablon:Portál/leírások"]},"scribunto":{"limitreport-timeusage":{"value":"0.043","limit":"10.000"},"limitreport-memusage":{"value":639566,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-7fc47fc68d-qrqh9","timestamp":"20241128185746","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Naiv halmazelm\u00e9let","url":"https:\/\/hu.wikipedia.org\/wiki\/Naiv_halmazelm%C3%A9let","sameAs":"http:\/\/www.wikidata.org\/entity\/Q903783","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q903783","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-07-10T19:43:43Z"}</script> </body> </html>