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id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E9%9B%86%E5%90%88%E8%AE%BA" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E9%9B%86%E5%90%88%E8%AE%BA" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</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"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</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/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</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> </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="站点"> <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="目录" 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">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</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">序言</div> </a> </li> <li id="toc-歷史" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#歷史"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>歷史</span> </div> </a> <ul id="toc-歷史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-基礎概念及符號" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#基礎概念及符號"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>基礎概念及符號</span> </div> </a> <ul id="toc-基礎概念及符號-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-集合的本體論" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#集合的本體論"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>集合的本體論</span> </div> </a> <ul id="toc-集合的本體論-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-公理集合論" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#公理集合論"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>公理集合論</span> </div> </a> <ul id="toc-公理集合論-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-應用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#應用"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>應用</span> </div> </a> <ul id="toc-應用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-研究領域" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#研究領域"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>研究領域</span> </div> </a> <button aria-controls="toc-研究領域-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>开关研究領域子章节</span> </button> <ul id="toc-研究領域-sublist" class="vector-toc-list"> <li id="toc-組合集合論" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#組合集合論"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>組合集合論</span> </div> </a> <ul id="toc-組合集合論-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-描述集合論" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#描述集合論"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>描述集合論</span> </div> </a> <ul id="toc-描述集合論-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-大基數" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#大基數"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>大基數</span> </div> </a> <ul id="toc-大基數-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-模糊集" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#模糊集"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>模糊集</span> </div> </a> <ul id="toc-模糊集-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-力迫" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#力迫"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>力迫</span> </div> </a> <ul id="toc-力迫-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-對集合論的異議" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#對集合論的異議"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>對集合論的異議</span> </div> </a> <ul id="toc-對集合論的異議-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-相關條目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#相關條目"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>相關條目</span> </div> </a> <ul id="toc-相關條目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考資料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考資料"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>參考資料</span> </div> </a> <ul id="toc-參考資料-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-額外閱讀" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#額外閱讀"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>額外閱讀</span> </div> </a> <ul id="toc-額外閱讀-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-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="目录" 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="开关目录" > <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">开关目录</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">集合论</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="前往另一种语言写成的文章。118种语言可用" > <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-118" 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">118种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Stelteorie" title="Stelteorie – 南非荷兰语" lang="af" hreflang="af" data-title="Stelteorie" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Mengenlehre" title="Mengenlehre – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Mengenlehre" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%A5%E1%8A%90_%E1%88%B5%E1%89%A5%E1%88%B5%E1%89%A5" title="ሥነ ስብስብ – 阿姆哈拉语" lang="am" hreflang="am" data-title="ሥነ ስብስብ" data-language-autonym="አማርኛ" data-language-local-name="阿姆哈拉语" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Teor%C3%ADa_de_conchuntos" title="Teoría de conchuntos – 阿拉贡语" lang="an" hreflang="an" data-title="Teoría de conchuntos" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%81%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%AF_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="समुच्चय सिद्धान्त – 昂加语" lang="anp" hreflang="anp" data-title="समुच्चय सिद्धान्त" data-language-autonym="अंगिका" data-language-local-name="昂加语" class="interlanguage-link-target"><span>अंगिका</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" title="نظرية المجموعات – 阿拉伯语" lang="ar" hreflang="ar" data-title="نظرية المجموعات" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A6%82%E0%A6%B9%E0%A6%A4%E0%A6%BF_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="সংহতি তত্ত্ব – 阿萨姆语" lang="as" hreflang="as" data-title="সংহতি তত্ত্ব" data-language-autonym="অসমীয়া" data-language-local-name="阿萨姆语" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_conxuntos" title="Teoría de conxuntos – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Teoría de conxuntos" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87oxluqlar_n%C9%99z%C9%99riyy%C9%99si" title="Çoxluqlar nəzəriyyəsi – 阿塞拜疆语" lang="az" hreflang="az" data-title="Çoxluqlar nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA%D1%82%D3%99%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Күмәклектәр теорияһы – 巴什基尔语" lang="ba" hreflang="ba" data-title="Күмәклектәр теорияһы" data-language-autonym="Башҡортса" data-language-local-name="巴什基尔语" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Aibiu_teuor%C4%97j%C4%97" title="Aibiu teuorėjė – 薩莫吉希亞文" lang="sgs" hreflang="sgs" data-title="Aibiu teuorėjė" data-language-autonym="Žemaitėška" data-language-local-name="薩莫吉希亞文" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Teorya_nin_set" title="Teorya nin set – Central Bikol" lang="bcl" hreflang="bcl" data-title="Teorya nin set" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0%D1%9E" title="Тэорыя мностваў – 白俄罗斯语" lang="be" hreflang="be" data-title="Тэорыя мностваў" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0%D1%9E" title="Тэорыя мностваў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэорыя мностваў" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%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="Теория на множествата – 保加利亚语" lang="bg" hreflang="bg" data-title="Теория на множествата" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%87%E0%A6%9F_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="সেট তত্ত্ব – 孟加拉语" lang="bn" hreflang="bn" data-title="সেট তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Teorienn_an_teskado%C3%B9" title="Teorienn an teskadoù – 布列塔尼语" lang="br" hreflang="br" data-title="Teorienn an teskadoù" data-language-autonym="Brezhoneg" data-language-local-name="布列塔尼语" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Teorija_skupova" title="Teorija skupova – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Teorija skupova" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_conjunts" title="Teoria de conjunts – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Teoria de conjunts" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%DA%A9%DB%86%D9%85%DB%95%DA%B5%DB%95" title="تیۆریی کۆمەڵە – 中库尔德语" lang="ckb" hreflang="ckb" data-title="تیۆریی کۆمەڵە" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_mno%C5%BEin" title="Teorie množin – 捷克语" lang="cs" hreflang="cs" data-title="Teorie množin" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%99%D1%8B%D1%88%D1%81%D0%B5%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Йышсен теорийĕ – 楚瓦什语" lang="cv" hreflang="cv" data-title="Йышсен теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Damcaniaeth_setiau" title="Damcaniaeth setiau – 威尔士语" lang="cy" hreflang="cy" data-title="Damcaniaeth setiau" data-language-autonym="Cymraeg" data-language-local-name="威尔士语" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/M%C3%A6ngdel%C3%A6re" title="Mængdelære – 丹麦语" lang="da" hreflang="da" data-title="Mængdelære" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mengenlehre" title="Mengenlehre – 德语" lang="de" hreflang="de" data-title="Mengenlehre" data-language-autonym="Deutsch" data-language-local-name="德语" 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%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%83%CF%85%CE%BD%CF%8C%CE%BB%CF%89%CE%BD" title="Θεωρία συνόλων – 希腊语" lang="el" hreflang="el" data-title="Θεωρία συνόλων" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Set_theory" title="Set theory – 英语" lang="en" hreflang="en" data-title="Set theory" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Aro-teorio" title="Aro-teorio – 世界语" lang="eo" hreflang="eo" data-title="Aro-teorio" data-language-autonym="Esperanto" data-language-local-name="世界语" 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_de_conjuntos" title="Teoría de conjuntos – 西班牙语" lang="es" hreflang="es" data-title="Teoría de conjuntos" data-language-autonym="Español" data-language-local-name="西班牙语" 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/Hulgateooria" title="Hulgateooria – 爱沙尼亚语" lang="et" hreflang="et" data-title="Hulgateooria" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" 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/Multzo-teoria" title="Multzo-teoria – 巴斯克语" lang="eu" hreflang="eu" data-title="Multzo-teoria" data-language-autonym="Euskara" data-language-local-name="巴斯克语" 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_%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87%E2%80%8C%D9%87%D8%A7" title="نظریه مجموعه‌ها – 波斯语" lang="fa" hreflang="fa" data-title="نظریه مجموعه‌ها" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Joukko-oppi" title="Joukko-oppi – 芬兰语" lang="fi" hreflang="fi" data-title="Joukko-oppi" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Hulgateooria" title="Hulgateooria – 佛羅文" lang="vro" hreflang="vro" data-title="Hulgateooria" data-language-autonym="Võro" data-language-local-name="佛羅文" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Mongdarl%C3%A6ra" title="Mongdarlæra – 法罗语" lang="fo" hreflang="fo" data-title="Mongdarlæra" data-language-autonym="Føroyskt" data-language-local-name="法罗语" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_ensembles" title="Théorie des ensembles – 法语" lang="fr" hreflang="fr" data-title="Théorie des ensembles" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Mengdeliar" title="Mengdeliar – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Mengdeliar" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Teorie_dai_insiemis" title="Teorie dai insiemis – 弗留利语" lang="fur" hreflang="fur" data-title="Teorie dai insiemis" data-language-autonym="Furlan" data-language-local-name="弗留利语" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tacartheoiric" title="Tacartheoiric – 爱尔兰语" lang="ga" hreflang="ga" data-title="Tacartheoiric" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_conxuntos" title="Teoría de conxuntos – 加利西亚语" lang="gl" hreflang="gl" data-title="Teoría de conxuntos" data-language-autonym="Galego" data-language-local-name="加利西亚语" 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" title="תורת הקבוצות – 希伯来语" lang="he" hreflang="he" data-title="תורת הקבוצות" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%81%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%AF_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="समुच्चय सिद्धान्त – 印地语" lang="hi" hreflang="hi" data-title="समुच्चय सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Set_theory" title="Set theory – 斐濟印地文" lang="hif" hreflang="hif" data-title="Set theory" data-language-autonym="Fiji Hindi" data-language-local-name="斐濟印地文" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_skupova" title="Teorija skupova – 克罗地亚语" lang="hr" hreflang="hr" data-title="Teorija skupova" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Halmazelm%C3%A9let" title="Halmazelmélet – 匈牙利语" lang="hu" hreflang="hu" data-title="Halmazelmélet" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%A6%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բազմությունների տեսություն – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Բազմությունների տեսություն" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Theoria_de_insimules" title="Theoria de insimules – 国际语" lang="ia" hreflang="ia" data-title="Theoria de insimules" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_himpunan" title="Teori himpunan – 印度尼西亚语" lang="id" hreflang="id" data-title="Teori himpunan" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Ensemblo-teorio" title="Ensemblo-teorio – 伊多语" lang="io" hreflang="io" data-title="Ensemblo-teorio" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Mengjafr%C3%A6%C3%B0i" title="Mengjafræði – 冰岛语" lang="is" hreflang="is" data-title="Mengjafræði" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_degli_insiemi" title="Teoria degli insiemi – 意大利语" lang="it" hreflang="it" data-title="Teoria degli insiemi" data-language-autonym="Italiano" data-language-local-name="意大利语" 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/%E9%9B%86%E5%90%88%E8%AB%96" title="集合論 – 日语" lang="ja" hreflang="ja" data-title="集合論" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/T%C3%A9ori_impunan" title="Téori impunan – 爪哇语" lang="jv" hreflang="jv" data-title="Téori impunan" data-language-autonym="Jawa" data-language-local-name="爪哇语" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%9A%E1%83%94%E1%83%97%E1%83%90_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="სიმრავლეთა თეორია – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="სიმრავლეთა თეორია" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Kegbe%C9%A3lasi_l%C9%A9ma%C9%A3z%C9%A9y%C9%9B" title="Kegbeɣlasi lɩmaɣzɩyɛ – Kabiye" lang="kbp" hreflang="kbp" data-title="Kegbeɣlasi lɩmaɣzɩyɛ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Thiori_ya_Seti" title="Thiori ya Seti – 吉库尤语" lang="ki" hreflang="ki" data-title="Thiori ya Seti" data-language-autonym="Gĩkũyũ" data-language-local-name="吉库尤语" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D0%B8%D1%8B%D0%BD%D0%B4%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Жиындар теориясы – 哈萨克语" lang="kk" hreflang="kk" data-title="Жиындар теориясы" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3_%E0%B2%B8%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%A7%E0%B2%BE%E0%B2%82%E0%B2%A4" title="ಗಣ ಸಿದ್ಧಾಂತ – 卡纳达语" lang="kn" hreflang="kn" data-title="ಗಣ ಸಿದ್ಧಾಂತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="卡纳达语" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%91%ED%95%A9%EB%A1%A0" title="집합론 – 韩语" lang="ko" hreflang="ko" data-title="집합론" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D3%A9%D0%BF%D1%82%D2%AF%D0%BA%D1%82%D3%A9%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Көптүктөр теориясы – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Көптүктөр теориясы" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><a href="https://la.wikipedia.org/wiki/Theoria_copiarum" title="Theoria copiarum – 拉丁语" lang="la" hreflang="la" data-title="Theoria copiarum" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teur%C3%ADa_di_cungjuunt" title="Teuría di cungjuunt – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Teuría di cungjuunt" data-language-autonym="Lombard" data-language-local-name="倫巴底文" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Aibi%C5%B3_teorija" title="Aibių teorija – 立陶宛语" lang="lt" hreflang="lt" data-title="Aibių teorija" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kopu_teorija" title="Kopu teorija – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Kopu teorija" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%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="Теорија на множествата – 马其顿语" lang="mk" hreflang="mk" data-title="Теорија на множествата" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="ഗണസിദ്ധാന്തം – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="ഗണസിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%9A%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A4%BE%E0%A4%A6" title="संचप्रवाद – 马拉地语" lang="mr" hreflang="mr" data-title="संचप्रवाद" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_set" title="Teori set – 马来语" lang="ms" hreflang="ms" data-title="Teori set" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%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="အစုသီအိုရီ – 缅甸语" lang="my" hreflang="my" data-title="အစုသီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="缅甸语" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Mengdenleer" title="Mengdenleer – 低地德语" lang="nds" hreflang="nds" data-title="Mengdenleer" data-language-autonym="Plattdüütsch" data-language-local-name="低地德语" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B8%E0%A5%87%E0%A4%9F_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="सेट सिद्धान्त – 尼瓦尔语" lang="new" hreflang="new" data-title="सेट सिद्धान्त" data-language-autonym="नेपाल भाषा" data-language-local-name="尼瓦尔语" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Verzamelingenleer" title="Verzamelingenleer – 荷兰语" lang="nl" hreflang="nl" data-title="Verzamelingenleer" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Mengdel%C3%A6re" title="Mengdelære – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Mengdelære" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Mengdel%C3%A6re" title="Mengdelære – 书面挪威语" lang="nb" hreflang="nb" data-title="Mengdelære" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Ensemble-teorie" title="Ensemble-teorie – 諾維亞文" lang="nov" hreflang="nov" data-title="Ensemble-teorie" data-language-autonym="Novial" data-language-local-name="諾維亞文" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Teoria_deis_ensembles" title="Teoria deis ensembles – 奥克语" lang="oc" hreflang="oc" data-title="Teoria deis ensembles" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%B8%E0%AD%87%E0%AC%9F_%E0%AC%A4%E0%AC%A4%E0%AD%8D%E0%AC%A4%E0%AD%8D%E0%AD%B1" title="ସେଟ ତତ୍ତ୍ୱ – 奥里亚语" lang="or" hreflang="or" data-title="ସେଟ ତତ୍ତ୍ୱ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="奥里亚语" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A9%88%E0%A8%9F_%E0%A8%B8%E0%A8%BF%E0%A8%A7%E0%A8%BE%E0%A8%82%E0%A8%A4" title="ਸੈਟ ਸਿਧਾਂਤ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਸੈਟ ਸਿਧਾਂਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Mengenlehre" title="Mengenlehre – 普法爾茨德文" lang="pfl" hreflang="pfl" data-title="Mengenlehre" data-language-autonym="Pälzisch" data-language-local-name="普法爾茨德文" class="interlanguage-link-target"><span>Pälzisch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_mnogo%C5%9Bci" title="Teoria mnogości – 波兰语" lang="pl" hreflang="pl" data-title="Teoria mnogości" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teor%C3%ACa_dj%27ansem" title="Teorìa dj'ansem – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Teorìa dj'ansem" data-language-autonym="Piemontèis" data-language-local-name="皮埃蒙特文" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B3%DB%8C%D9%B9_%D8%AA%DA%BE%DB%8C%D9%88%D8%B1%DB%8C" title="سیٹ تھیوری – Western Punjabi" lang="pnb" hreflang="pnb" data-title="سیٹ تھیوری" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D8%B3%D9%8A%D9%BC_%D9%86%D8%B8%D8%B1%D9%8A%D9%87" title="د سيټ نظريه – 普什图语" lang="ps" hreflang="ps" data-title="د سيټ نظريه" data-language-autonym="پښتو" data-language-local-name="普什图语" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_dos_conjuntos" title="Teoria dos conjuntos – 葡萄牙语" lang="pt" hreflang="pt" data-title="Teoria dos conjuntos" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_mul%C8%9Bimilor" title="Teoria mulțimilor – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Teoria mulțimilor" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%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="Теория множеств – 俄语" lang="ru" hreflang="ru" data-title="Теория множеств" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD" title="Теорія множин – 盧森尼亞文" lang="rue" hreflang="rue" data-title="Теорія множин" data-language-autonym="Русиньскый" data-language-local-name="盧森尼亞文" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A2%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="Теория множеств – 萨哈语" lang="sah" hreflang="sah" data-title="Теория множеств" data-language-autonym="Саха тыла" data-language-local-name="萨哈语" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Set_theory" title="Set theory – 苏格兰语" lang="sco" hreflang="sco" data-title="Set theory" data-language-autonym="Scots" data-language-local-name="苏格兰语" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_skupova" title="Teorija skupova – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Teorija skupova" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%94%E0%B6%BD%E0%B6%9A_%E0%B7%80%E0%B7%8F%E0%B6%AF%E0%B6%BA" title="කුලක වාදය – 僧伽罗语" lang="si" hreflang="si" data-title="කුලක වාදය" data-language-autonym="සිංහල" data-language-local-name="僧伽罗语" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Set_theory" title="Set theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Te%C3%B3ria_mno%C5%BE%C3%ADn" title="Teória množín – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Teória množín" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teorija_mno%C5%BEic" title="Teorija množic – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Teorija množic" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teoria_e_vendosur" title="Teoria e vendosur – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Teoria e vendosur" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D1%81%D0%BA%D1%83%D0%BF%D0%BE%D0%B2%D0%B0" title="Теорија скупова – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Теорија скупова" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%A4ngdteori" title="Mängdteori – 瑞典语" lang="sv" hreflang="sv" data-title="Mängdteori" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Nadharia_seti" title="Nadharia seti – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Nadharia seti" data-language-autonym="Kiswahili" data-language-local-name="斯瓦希里语" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="கணக் கோட்பாடு – 泰米尔语" lang="ta" hreflang="ta" data-title="கணக் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B9%80%E0%B8%8B%E0%B8%95" title="ทฤษฎีเซต – 泰语" lang="th" hreflang="th" data-title="ทฤษฎีเซต" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/K%C3%B6pl%C3%BCkler_we_san_k%C3%B6pl%C3%BCkleri" title="Köplükler we san köplükleri – 土库曼语" lang="tk" hreflang="tk" data-title="Köplükler we san köplükleri" data-language-autonym="Türkmençe" data-language-local-name="土库曼语" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_pangkat" title="Teorya ng pangkat – 他加禄语" lang="tl" hreflang="tl" data-title="Teorya ng pangkat" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCmeler_teorisi" title="Kümeler teorisi – 土耳其语" lang="tr" hreflang="tr" data-title="Kümeler teorisi" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BF%D0%BB%D0%B5%D0%BA_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D0%B5" title="Күплек теориясе – 鞑靼语" lang="tt" hreflang="tt" data-title="Күплек теориясе" data-language-autonym="Татарча / tatarça" data-language-local-name="鞑靼语" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD" title="Теорія множин – 乌克兰语" lang="uk" hreflang="uk" data-title="Теорія множин" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%D8%B7%D8%A7%D9%82%D9%85" title="نظریۂ طاقم – 乌尔都语" lang="ur" hreflang="ur" data-title="نظریۂ طاقم" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/To%CA%BBplamlar_nazariyasi" title="Toʻplamlar nazariyasi – 乌兹别克语" lang="uz" hreflang="uz" data-title="Toʻplamlar nazariyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/%C3%84j%C3%BCziden_teorii" title="Äjüziden teorii – 维普森语" lang="vep" hreflang="vep" data-title="Äjüziden teorii" data-language-autonym="Vepsän kel’" data-language-local-name="维普森语" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_t%E1%BA%ADp_h%E1%BB%A3p" title="Lý thuyết tập hợp – 越南语" lang="vi" hreflang="vi" data-title="Lý thuyết tập hợp" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Verzoameliengeleer" title="Verzoameliengeleer – 西佛蘭德文" lang="vls" hreflang="vls" data-title="Verzoameliengeleer" data-language-autonym="West-Vlams" data-language-local-name="西佛蘭德文" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-vo mw-list-item"><a href="https://vo.wikipedia.org/wiki/Konletateor" title="Konletateor – 沃拉普克语" lang="vo" hreflang="vo" data-title="Konletateor" data-language-autonym="Volapük" data-language-local-name="沃拉普克语" class="interlanguage-link-target"><span>Volapük</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Teyorya_set" title="Teyorya set – 瓦瑞语" lang="war" hreflang="war" data-title="Teyorya set" data-language-autonym="Winaray" data-language-local-name="瓦瑞语" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论 – 吴语" lang="wuu" hreflang="wuu" data-title="集合论" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9B%E1%83%98%E1%83%90%E1%83%A0%E1%83%94%E1%83%94%E1%83%A4%E1%83%98%E1%83%A8_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="მიარეეფიშ თეორია – 明格列爾文" lang="xmf" hreflang="xmf" data-title="მიარეეფიშ თეორია" data-language-autonym="მარგალური" data-language-local-name="明格列爾文" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%A2%D7%96%D7%A2%D7%9E%D7%9C%D7%A2%D7%9F_%D7%98%D7%A2%D7%90%D7%A8%D7%99%D7%A2" title="געזעמלען טעאריע – 意第绪语" lang="yi" hreflang="yi" data-title="געזעמלען טעאריע" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%9B%86%E8%AB%96" title="集論 – 文言文" lang="lzh" hreflang="lzh" data-title="集論" data-language-autonym="文言" data-language-local-name="文言文" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Chi%CC%8Dp-ha%CC%8Dp-l%C5%ABn" title="Chi̍p-ha̍p-lūn – 闽南语" lang="nan" hreflang="nan" data-title="Chi̍p-ha̍p-lūn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="闽南语" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%86%E5%90%88%E8%AB%96" title="集合論 – 粤语" lang="yue" hreflang="yue" data-title="集合論" data-language-autonym="粵語" data-language-local-name="粤语" 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/Q12482#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</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="命名空间"> <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 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lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E9%9B%86%E5%90%88%E8%AE%BA" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E9%9B%86%E5%90%88%E8%AE%BA" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E9%9B%86%E5%90%88%E8%AE%BA" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <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 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cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">工具</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">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </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/%E9%9B%86%E5%90%88%E8%AE%BA"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> 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//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><div id="noteTA-8215fff0" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div><div class="noteTA-local"><div data-noteta-code="zh-hans:对象; zh-hant:物件;"></div></div></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Venn_A_intersect_B.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/220px-Venn_A_intersect_B.svg.png" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/330px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/440px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a><figcaption>二個<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a><a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>的<a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏圖</a></figcaption></figure> <p><b>集合論</b>（英語：<span lang="en">set theory</span>）或稱<b>集論</b>，是研究<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>（由一堆<a href="/wiki/%E6%8A%BD%E8%B1%A1%E5%AF%B9%E8%B1%A1" class="mw-redirect" title="抽象对象">抽象对象</a>構成的整體）的<a href="/wiki/%E6%95%B8%E5%AD%B8" class="mw-redirect" title="數學">數學</a>理論，包含集合和<a href="/wiki/%E5%85%83%E7%B4%A0_(%E6%95%B8%E5%AD%B8)" title="元素 (數學)">元素</a>（或稱為<a href="/wiki/%E6%88%90%E5%93%A1_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="成員 (數學)">成員</a>）、<a href="/wiki/%E5%85%B3%E7%B3%BB_(%E6%95%B0%E5%AD%A6)" title="关系 (数学)">關係</a>等最基本數學<a href="/wiki/%E6%A6%82%E5%BF%B5" title="概念">概念</a>。在大多數現代數學的公式化中，都是在集合論的語言下談論各種<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AF%B9%E8%B1%A1" title="数学对象">数学对象</a>。集合論、<a href="/wiki/%E5%91%BD%E9%A2%98%E9%80%BB%E8%BE%91" title="命题逻辑">命題邏輯</a>與<a href="/wiki/%E8%B0%93%E8%AF%8D%E9%80%BB%E8%BE%91" title="谓词逻辑">謂詞邏輯</a>共同構成了數學的<a href="/wiki/%E6%95%B8%E5%AD%B8%E5%9F%BA%E7%A4%8E" class="mw-redirect" title="數學基礎">公理化基礎</a>，以未定義的「集合」與「集合成員」等術語來形式化地建構<a href="/wiki/%E6%95%B8%E5%AD%B8%E7%89%A9%E4%BB%B6" class="mw-redirect" title="數學物件">數學物件</a>。 </p><p>現代集合論的研究是在1870年代由<a href="/wiki/%E4%BF%84%E5%9B%BD" class="mw-redirect" title="俄国">俄国</a>数学家<a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94" class="mw-redirect" title="康托尔">康托爾</a>及<a href="/wiki/%E5%BE%B7%E5%9C%8B" class="mw-redirect" title="德國">德國</a>数学家<a href="/wiki/%E7%90%86%E5%AF%9F%C2%B7%E6%88%B4%E5%BE%B7%E9%87%91" class="mw-redirect" title="理察·戴德金">理察·戴德金</a>的<a href="/wiki/%E6%9C%B4%E7%B4%A0%E9%9B%86%E5%90%88%E8%AE%BA" title="朴素集合论">樸素集合論</a>開始。在樸素集合論中，集合是當做一堆物件構成的整體之類的自證概念，沒有有關集合的形式化定義。在發現樸素集合論會產生一些<span class="ilh-all" data-orig-title="集合論的悖論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Paradoxes of set theory"><span class="ilh-page"><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AB%96%E7%9A%84%E6%82%96%E8%AB%96&action=edit&redlink=1" class="new" title="集合論的悖論（页面不存在）">悖論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Paradoxes_of_set_theory" class="extiw" title="en:Paradoxes of set theory"><span lang="en" dir="auto">Paradoxes of set theory</span></a></span>）</span></span>後，二十世紀初期提出了許多<a href="/wiki/%E5%85%AC%E7%90%86%E5%8C%96%E9%9B%86%E5%90%88%E8%AB%96" class="mw-redirect" title="公理化集合論">公理化集合論</a>，其中最著名的是包括<a href="/wiki/%E9%81%B8%E6%93%87%E5%85%AC%E7%90%86" class="mw-redirect" title="選擇公理">選擇公理</a>的<a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅洛-弗蘭克爾集合論</a>，簡稱ZFC。公理化集合論不直接定義集合和集合成員，而是先規範可以描述其性質的一些<a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a>。 </p><p>集合論常被視為<a href="/wiki/%E6%95%B8%E5%AD%B8%E5%9F%BA%E7%A4%8E" class="mw-redirect" title="數學基礎">數學基礎</a>之一，特別是 ZFC 集合論。除了其基礎的作用外，集合論也是數學理論中的一部份，當代的集合論研究有許多離散的主題，從<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>線的結構到<a href="/wiki/%E5%A4%A7%E5%9F%BA%E6%95%B0" title="大基数">大基数</a>的<a href="/wiki/%E4%B8%80%E8%87%B4%E6%80%A7_(%E9%82%8F%E8%BC%AF)" title="一致性 (邏輯)">一致性</a>等。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="歷史"><span id=".E6.AD.B7.E5.8F.B2"></span>歷史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=1" title="编辑章节：歷史"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Georg_Cantor_1894.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Georg_Cantor_1894.jpg/160px-Georg_Cantor_1894.jpg" decoding="async" width="160" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Georg_Cantor_1894.jpg/240px-Georg_Cantor_1894.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/73/Georg_Cantor_1894.jpg 2x" data-file-width="271" data-file-height="389" /></a><figcaption>康托爾</figcaption></figure> <p>現代集合論的研究開始於1870年代由<a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94" class="mw-redirect" title="康托尔">康托爾</a>及<a href="/wiki/%E7%90%86%E5%AF%9F%C2%B7%E6%88%B4%E5%BE%B7%E9%87%91" class="mw-redirect" title="理察·戴德金">理察·戴德金</a>提出的<a href="/wiki/%E6%9C%B4%E7%B4%A0%E9%9B%86%E5%90%88%E8%AE%BA" title="朴素集合论">樸素集合論</a>。一般數學主題的出現及發展都是由多名研究者的互動中產生的，但樸素集合論的開始是1874年康托爾的一篇論文《On a Characteristic Property of All Real Algebraic Numbers》<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。而在稍早的1873年12月7日，康托尔写信给戴德金，说他已能成功地证明实数的“集体”是不可数的了，这一天也因此成为了<a class="mw-selflink selflink">集合论</a>的诞生日。 </p><p>從西元前五世紀時，數學家們就在研究有關<a href="/wiki/%E7%84%A1%E7%AA%AE" class="mw-redirect" title="無窮">無窮</a>的性質，最早期是<a href="/wiki/%E5%B8%8C%E8%87%98" class="mw-redirect" title="希臘">希臘</a>數學家<a href="/wiki/%E5%9F%83%E5%88%A9%E4%BA%9A%E7%9A%84%E8%8A%9D%E8%AF%BA" title="埃利亚的芝诺">芝諾</a>和印度數學家，十九世紀時<a href="/wiki/%E4%BC%AF%E7%B4%8D%E5%BE%B7%C2%B7%E6%B3%A2%E7%88%BE%E6%9F%A5%E8%AB%BE" class="mw-redirect" title="伯納德·波爾查諾">伯納德·波爾查諾</a>在此領域有相當的進展<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。現在對於無限的了解是從1867–71年康托爾在數論上的研究開始，1872年康托爾和<a href="/wiki/%E7%90%86%E6%9F%A5%E5%BE%B7%C2%B7%E6%88%B4%E5%BE%B7%E9%87%91" title="理查德·戴德金">理查德·戴德金</a>的一次聚會影響了康托爾的理念，最後產生了1874年的論文。 </p><p>當時的數學家對康托爾的研究有二種完全不同的反應：<a href="/wiki/%E5%8D%A1%E5%B0%94%C2%B7%E9%AD%8F%E5%B0%94%E6%96%AF%E7%89%B9%E6%8B%89%E6%96%AF" class="mw-redirect" title="卡尔·魏尔斯特拉斯">卡尔·魏尔斯特拉斯</a>及理查德·戴德金支持康托爾的研究，而像<a href="/wiki/%E5%88%A9%E5%A5%A5%E6%B3%A2%E5%BE%B7%C2%B7%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B" title="利奥波德·克罗内克">利奥波德·克罗内克</a>等<a href="/wiki/%E6%95%B0%E5%AD%A6%E7%BB%93%E6%9E%84%E4%B8%BB%E4%B9%89" class="mw-redirect" title="数学结构主义">结构主义者</a>則持反對態度。康托爾的研究後來廣為流傳，原因是當中概念的實效性，例如集合之間的<a href="/wiki/%E5%8F%8C%E5%B0%84" title="双射">双射</a>，康托爾對於<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>較整數多的證明，以及由<a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">冪集</a>所產生「無窮的無窮」的概念，等等。這些概念最後成為1898年《<span class="ilh-all" data-orig-title="克莱因的百科全书" data-lang-code="en" data-lang-name="英语" data-foreign-title="Klein's encyclopedia"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%8B%E8%8E%B1%E5%9B%A0%E7%9A%84%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6&action=edit&redlink=1" class="new" title="克莱因的百科全书（页面不存在）">克莱因的百科全书</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Klein%27s_encyclopedia" class="extiw" title="en:Klein's encyclopedia"><span lang="en" dir="auto">Klein's encyclopedia</span></a></span>）</span></span>》中的《Mengenlehre》（集合論）條目。 </p><p>在1900年左右許多數學家發現樸素集合論會產生一些矛盾的情形，稱為<a href="/wiki/%E4%BA%8C%E5%BE%8B%E8%83%8C%E5%8F%8D" title="二律背反">二律背反</a>或是<a href="/wiki/%E6%82%96%E8%AE%BA" title="悖论">悖论</a>，<a href="/wiki/%E4%BC%AF%E7%89%B9%E5%85%B0%C2%B7%E7%BD%97%E7%B4%A0" title="伯特兰·罗素">伯特兰·罗素</a>和<a href="/wiki/%E6%81%A9%E6%96%AF%E7%89%B9%C2%B7%E7%AD%96%E6%A2%85%E6%B4%9B" title="恩斯特·策梅洛">恩斯特·策梅洛</a>均發現了最簡單的悖论，也就是現在所稱的<a href="/wiki/%E7%BD%97%E7%B4%A0%E6%82%96%E8%AE%BA" title="罗素悖论">罗素悖论</a>：考慮「由所有不包含集合自身的集合所構成的集合」，記之為 <i>S</i>。不管假設 <i>S</i> 是或不是 <i>S</i> 自身的元素，按照 <i>S</i> 的定義都會導致矛盾。罗素悖论也造成了第三次數學危機。 </p><p>1899年時康托爾自己也提出一個會產生悖论的問題「一個由所有集合形成的集合，其<a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="基数 (数学)">基數</a>為何？」因而產生<a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94%E6%82%96%E8%AE%BA" title="康托尔悖论">康托尔悖论</a>。羅素在1903年他所著的《<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%8E%9F%E7%90%86" title="数学原理">数学原理</a>》中也用此悖論來評論當時的歐陸數學。 </p><p>不過上述的爭論沒有使數學家放棄集合論，<a href="/wiki/%E6%81%A9%E6%96%AF%E7%89%B9%C2%B7%E7%AD%96%E6%A2%85%E6%B4%9B" title="恩斯特·策梅洛">恩斯特·策梅洛</a>及<a href="/wiki/%E4%BA%9A%E4%BC%AF%E6%8B%89%E7%BD%95%C2%B7%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94" title="亚伯拉罕·弗兰克尔">亚伯拉罕·弗兰克尔</a>分別在1908年和1922年的研究．最後產生了<a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅洛-弗兰克尔集合论</a>的許多公理。<a href="/wiki/%E6%98%82%E5%88%A9%C2%B7%E5%8B%92%E8%B2%9D%E6%A0%BC" class="mw-redirect" title="昂利·勒貝格">昂利·勒貝格</a>等人在<a href="/wiki/%E5%AF%A6%E5%88%86%E6%9E%90" class="mw-redirect" title="實分析">實分析</a>上的研究用到集合論中的許多數學工具，後來集合論也成為近代數學的一部份。集合論已被視為是數學的基礎理論，不過在一些領域中<a href="/wiki/%E8%8C%83%E7%95%B4%E8%AE%BA" title="范畴论">范畴论</a>被認為是更適合的基礎理論。 </p> <div class="mw-heading mw-heading2"><h2 id="基礎概念及符號"><span id=".E5.9F.BA.E7.A4.8E.E6.A6.82.E5.BF.B5.E5.8F.8A.E7.AC.A6.E8.99.9F"></span>基礎概念及符號</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=2" title="编辑章节：基礎概念及符號"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合 (数学)</a>和<a href="/wiki/%E9%9B%86%E5%90%88%E4%BB%A3%E6%95%B8" class="mw-redirect" title="集合代數">集合代數</a></div> <p>集合论是從一個对象<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span>和<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</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 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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>之間的<a href="/wiki/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" title="二元关系">二元关系</a>開始：若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span>是<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>的<a href="/wiki/%E5%85%83%E7%B4%A0_(%E6%95%B8%E5%AD%B8)" title="元素 (數學)">元素</a>，可表示為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109f5d49e1ba1bcc71593b98c7f51e613ddaa216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.711ex; height:2.176ex;" alt="{\displaystyle o\in A}"></span>。由於集合也是一個对象，因此上述關係也可以用在集合和集合的關係。 </p><p>另外一種兩個集合之間的關係，稱為包含關係。若集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>中的所有元素都是集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>中的元素，則稱集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的<a href="/wiki/%E5%AD%90%E9%9B%86" title="子集">子集</a>，符號為<span class="mwe-math-element"><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\subseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}"></span>。例如<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111aef9df2c1a71ed7820a5426ef3e8c7b781564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{1,2\}}"></span>是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span>的子集，但<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489752080d5ab14fdd4b9900c868aa9ad75b1193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{1,4\}}"></span>就不是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span>的子集。依照定義，任一個集合也是本身的子集，不考慮本身的子集稱為真子集。集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>為集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的真子集<a href="/wiki/%E8%8B%A5%E4%B8%94%E5%94%AF%E8%8B%A5" class="mw-redirect" title="若且唯若">若且唯若</a>集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>為集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的子集，且集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>不是集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>的子集。 </p><p><a href="/wiki/%E6%95%B0" title="数">数</a>的<a href="/wiki/%E7%AE%97%E8%A1%93" class="mw-redirect" title="算術">算術</a>中有許多一元及二元运算，集合論也有許多針對集合的一元及二元运算： </p> <ul><li>集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的<a href="/wiki/%E8%81%AF%E9%9B%86" class="mw-redirect" title="聯集">聯集</a>，符號為<span class="mwe-math-element"><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\cup B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cup B}"></span>，是在至少在集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>或<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>中出現的元素，集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span>和集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99f7ef354627214c29c9a97a6035208056f934a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{2,3,4\}}"></span>的聯集為集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ca66fd59843b349aed8ffa7655c1aae77625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.077ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,4\}}"></span>。</li> <li>集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>，符號為<span class="mwe-math-element"><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\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb27b38cf9eac6060e67b61f66cd9beec5067f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cap B}"></span>，是同時在集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>及<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>中出現的元素，集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span>和集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99f7ef354627214c29c9a97a6035208056f934a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{2,3,4\}}"></span>的交集為集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d118b286f8bcc9dad1b4f5f37d5a7342d95f2e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{2,3\}}"></span>。</li> <li>集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>和<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>的<a href="/wiki/%E8%A1%A5%E9%9B%86" title="补集">相對差集</a>，符號為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\backslash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\backslash A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76206102d1ebbf8c8ac534ef4bb8599c7bd67a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.688ex; height:2.843ex;" alt="{\displaystyle U\backslash A}"></span>，是在集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>中，但不在集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>中的所有元素，相對差集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}\backslash \{2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}\backslash \{2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd1472a1c3bf3de6eda2697114c74bdbe1ea56f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.923ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}\backslash \{2,3,4\}}"></span>為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}"></span>，而相對差集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3,4\}\backslash \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,3,4\}\backslash \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ba8171051ca361c5710776a46fd3fae71bd994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.923ex; height:2.843ex;" alt="{\displaystyle \{2,3,4\}\backslash \{1,2,3\}}"></span>為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/247f12f7b428399a8b38022218e5d62f546af96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {4}}"></span>。當集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>是集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>的子集時，相對差集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\backslash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\backslash A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76206102d1ebbf8c8ac534ef4bb8599c7bd67a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.688ex; height:2.843ex;" alt="{\displaystyle U\backslash A}"></span>也稱為集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>在集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>中的<a href="/wiki/%E8%A3%9C%E9%9B%86" class="mw-redirect" title="補集">補集</a>。若是研究<a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏圖</a>，集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>為<a href="/wiki/%E5%85%A8%E9%9B%86" title="全集">全集</a>，且可以藉由上下文找到全集定義時，會使用<span class="mwe-math-element"><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^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7815638370f616885fa8162b06697d078459f5ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.687ex; height:2.343ex;" alt="{\displaystyle A^{c}}"></span>來代替<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\backslash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\backslash A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76206102d1ebbf8c8ac534ef4bb8599c7bd67a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.688ex; height:2.843ex;" alt="{\displaystyle U\backslash A}"></span>。</li> <li>集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的<a href="/wiki/%E5%AF%B9%E7%A7%B0%E5%B7%AE" title="对称差">对称差</a>，符號為<span class="mwe-math-element"><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\triangle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">△</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\triangle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b45893c234b0c54e9ef21deba20df2edd2292d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.573ex; height:2.176ex;" alt="{\displaystyle A\triangle B}"></span>或<span class="mwe-math-element"><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\ominus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊖</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ominus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd5c8d3fdf145dc31b779da31b1d8793d4fc059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\ominus B}"></span>，是指只在集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>及<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>中的其中一個出現，沒有在其交集中出現的元素。例如集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99f7ef354627214c29c9a97a6035208056f934a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{2,3,4\}}"></span>的对称差為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489752080d5ab14fdd4b9900c868aa9ad75b1193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{1,4\}}"></span>，也是其聯集和交集的相對差集<span class="mwe-math-element"><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\cup B)\backslash (A\cap B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\cup B)\backslash (A\cap B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4ff8bff35e8acf2271169e322c3a2529f4e904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.96ex; height:2.843ex;" alt="{\displaystyle (A\cup B)\backslash (A\cap B)}"></span>，或是二個相對差集的聯集<span class="mwe-math-element"><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\backslash B)\cup (B\backslash A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\backslash B)\cup (B\backslash A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608aa19f08d716119dad980e707e705c0204ed64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.54ex; height:2.843ex;" alt="{\displaystyle (A\backslash B)\cup (B\backslash A)}"></span>。</li> <li>集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的<a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%84%BF%E7%A7%AF" title="笛卡儿积">笛卡儿积</a>，符號為<span class="mwe-math-element"><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\times B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f31ae45b0098f06b5d22c38d317eb097a88fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle A\times B}"></span>，是一個由所有可能的<a href="/wiki/%E6%9C%89%E5%BA%8F%E5%AF%B9" title="有序对">有序对</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 (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span>形成的集合，其中第一个物件是<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>的成员，第二个物件是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>的成员。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111aef9df2c1a71ed7820a5426ef3e8c7b781564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{1,2\}}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\text{red}},{\text{white}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>red</mtext> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>white</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\text{red}},{\text{white}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e766d746511cf764b83b6fbd47567591309c892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.15ex; height:2.843ex;" alt="{\displaystyle \{{\text{red}},{\text{white}}\}}"></span>的笛卡儿积為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(1,{\text{red}}),(1,{\text{white}}),(2,{\text{red}}),(2,{\text{white}})\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>red</mtext> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>white</mtext> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>red</mtext> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>white</mtext> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,{\text{red}}),(1,{\text{white}}),(2,{\text{red}}),(2,{\text{white}})\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106f164a67a688f9fa17668122ad663507272138" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.031ex; height:2.843ex;" alt="{\displaystyle \{(1,{\text{red}}),(1,{\text{white}}),(2,{\text{red}}),(2,{\text{white}})\}}"></span>。</li> <li>集合<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>的<a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">冪集</a>是指以<span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>的全部子集為元素的集合，例如集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111aef9df2c1a71ed7820a5426ef3e8c7b781564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{1,2\}}"></span>的冪集為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varnothing ,\{1\},\{2\},\{1,2\}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\varnothing ,\{1\},\{2\},\{1,2\}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1529ac351bc1c605fa7ff73b75b6221c8232e66e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.893ex; height:2.843ex;" alt="{\displaystyle \{\varnothing ,\{1\},\{2\},\{1,2\}\}}"></span>。</li></ul> <p>一些重要的基本集合包括<a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a>（唯一沒有元素的集合），<a href="/wiki/%E6%95%B4%E6%95%B8" class="mw-redirect" title="整數">整數</a>集合及<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>集合。其他有關初等集合論的基本介紹，請參考<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="集合的本體論"><span id=".E9.9B.86.E5.90.88.E7.9A.84.E6.9C.AC.E9.AB.94.E8.AB.96"></span>集合的本體論</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=3" title="编辑章节：集合的本體論"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%86%AF%C2%B7%E8%AF%BA%E4%BC%8A%E6%9B%BC%E5%85%A8%E9%9B%86" title="冯·诺伊曼全集">冯·诺伊曼全集</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Von_Neumann_Hierarchy.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Von_Neumann_Hierarchy.svg/300px-Von_Neumann_Hierarchy.svg.png" decoding="async" width="300" height="310" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Von_Neumann_Hierarchy.svg/450px-Von_Neumann_Hierarchy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Von_Neumann_Hierarchy.svg/600px-Von_Neumann_Hierarchy.svg.png 2x" data-file-width="394" data-file-height="407" /></a><figcaption>冯·诺伊曼層次中的一部份</figcaption></figure> <p>若一個集合的所有元素都是集合，所有元素的元素都是集合……，此集合稱為<span class="ilh-all" data-orig-title="純集合" data-lang-code="en" data-lang-name="英语" data-foreign-title="pure set"><span class="ilh-page"><a href="/w/index.php?title=%E7%B4%94%E9%9B%86%E5%90%88&action=edit&redlink=1" class="new" title="純集合（页面不存在）">純集合</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/pure_set" class="extiw" title="en:pure set"><span lang="en" dir="auto">pure set</span></a></span>）</span></span>，例如只包括空集合的集合是一個非空的純集合。在當代的集合論中，常常嚴格限制只考慮純集合的<a href="/wiki/%E5%86%AF%C2%B7%E8%AF%BA%E4%BC%8A%E6%9B%BC%E5%85%A8%E9%9B%86" title="冯·诺伊曼全集">馮·諾伊曼全集</a>，許多公理集合論的系統也是為了純集合的公理化。這様的限制有許多技術上的優點，因為基本上所有的數學概念都可以用純集合來表示，上述的限制不影響相關的應用。馮·諾伊曼全集中的集合可以以累積層次（cumulative hierarchy）的方式整理，也就會依元素的深度、元素的元素的深度……來分類。層次中的每一個集合都會以<a href="/wiki/%E8%B6%85%E9%99%90%E5%BD%92%E7%BA%B3%E6%B3%95#超限递归" title="超限归纳法">超限递归</a>的方式指定一個<a href="/wiki/%E5%BA%8F%E6%95%B0" title="序数">序数</a>，稱為集合的階。純集合X的階定義為集合<span class="mwe-math-element"><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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>所有元素的階的<a href="/wiki/%E5%90%8E%E7%BB%A7%E5%BA%8F%E6%95%B0" title="后继序数">后继序数</a>的<a href="/wiki/%E6%9C%80%E5%B0%8F%E4%B8%8A%E7%95%8C" title="最小上界">最小上界</a>。例如空集的階定義為0，只包括空集的集合定義為1，針對每一個序数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>，集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582d40a9ff663187250948b07bb66456162c2042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle V_{\alpha }}"></span>按定義包含了所有階數小於<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>的純集合，整個馮·諾伊曼全集用<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>來表示。 </p> <div class="mw-heading mw-heading2"><h2 id="公理集合論"><span id=".E5.85.AC.E7.90.86.E9.9B.86.E5.90.88.E8.AB.96"></span>公理集合論</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=4" title="编辑章节：公理集合論"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>基礎集合論可以用非正式的、直覺的方式學習，在小學中就可以用<a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏圖</a>說明。基礎集合論直觀地假設集合就是一群符合任意特定條件的物件的組合，但此假設會造成<a href="/wiki/%E6%82%96%E8%AB%96" class="mw-redirect" title="悖論">悖論</a>。最簡單及著名的是<a href="/wiki/%E7%BE%85%E7%B4%A0%E6%82%96%E8%AB%96" class="mw-redirect" title="羅素悖論">羅素悖論</a>及<a href="/wiki/%E5%B8%83%E6%8B%89%E5%88%A9-%E7%A6%8F%E7%88%BE%E8%92%82%E6%82%96%E8%AB%96" class="mw-redirect" title="布拉利-福爾蒂悖論">布拉利-福爾蒂悖論</a>。<a href="/wiki/%E5%85%AC%E7%90%86%E9%9B%86%E5%90%88%E8%AB%96" class="mw-redirect" title="公理集合論">公理集合論</a>的形成就是為了避免這些集合論的悖論。 </p><p>許多數學家研究的公理集合論系統假設所有的集合形成累计层次。這類的系統可分為二類： </p> <ul><li>只由集合構成：這類系統包括最常用的公理集合論：含<a href="/wiki/%E9%81%B8%E6%93%87%E5%85%AC%E7%90%86" class="mw-redirect" title="選擇公理">選擇公理</a>的<a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅洛-弗兰克尔集合论</a>（ZFC），由<span class="ilh-all" data-orig-title="亞伯拉罕·弗蘭克爾" data-lang-code="en" data-lang-name="英语" data-foreign-title="Abraham Fraenkel"><span class="ilh-page"><a href="/wiki/%E4%BA%9A%E4%BC%AF%E6%8B%89%E7%BD%95%C2%B7%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94" title="亚伯拉罕·弗兰克尔">亞伯拉罕·弗蘭克爾</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Abraham_Fraenkel" class="extiw" title="en:Abraham Fraenkel"><span lang="en" dir="auto">Abraham Fraenkel</span></a></span>）</span></span>和<span class="ilh-all" data-orig-title="陶拉爾夫·斯科倫" data-lang-code="en" data-lang-name="英语" data-foreign-title="Thoralf Skolem"><span class="ilh-page"><a href="/w/index.php?title=%E9%99%B6%E6%8B%89%E7%88%BE%E5%A4%AB%C2%B7%E6%96%AF%E7%A7%91%E5%80%AB&action=edit&redlink=1" class="new" title="陶拉爾夫·斯科倫（页面不存在）">陶拉爾夫·斯科倫</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Thoralf_Skolem" class="extiw" title="en:Thoralf Skolem"><span lang="en" dir="auto">Thoralf Skolem</span></a></span>）</span></span>擴展了<a href="/wiki/%E7%AD%96%E6%A2%85%E7%BE%85%E9%9B%86%E5%90%88%E8%AB%96" class="mw-redirect" title="策梅羅集合論">策梅羅集合論</a>所得。其他和ZFC有關的集合論有： <ul><li><a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛集合论">策梅洛集合论</a>是由德國數學家<a href="/wiki/%E6%81%A9%E6%96%AF%E7%89%B9%C2%B7%E7%AD%96%E6%A2%85%E6%B4%9B" title="恩斯特·策梅洛">恩斯特·策梅洛</a>創立，將<a href="/wiki/%E5%88%86%E7%B1%BB%E5%85%AC%E7%90%86" title="分类公理">分类公理</a>代替<a href="/wiki/%E6%9B%BF%E4%BB%A3%E5%85%AC%E7%90%86" title="替代公理">替代公理</a>。</li> <li><span class="ilh-all" data-orig-title="廣義集合論" data-lang-code="en" data-lang-name="英语" data-foreign-title="General set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%BB%A3%E7%BE%A9%E9%9B%86%E5%90%88%E8%AB%96&action=edit&redlink=1" class="new" title="廣義集合論（页面不存在）">廣義集合論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/General_set_theory" class="extiw" title="en:General set theory"><span lang="en" dir="auto">General set theory</span></a></span>）</span></span>，策梅洛集合论的一小部份，已足以處理<a href="/wiki/%E7%9A%AE%E4%BA%9A%E8%AF%BA%E5%85%AC%E7%90%86" title="皮亚诺公理">皮亚诺公理</a>及<a href="/wiki/%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="有限集合">有限集合</a>。</li> <li><span class="ilh-all" data-orig-title="克里普克-普拉特克集理论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Kripke-Platek set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%8B%E9%87%8C%E6%99%AE%E5%85%8B-%E6%99%AE%E6%8B%89%E7%89%B9%E5%85%8B%E9%9B%86%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="克里普克-普拉特克集理论（页面不存在）">克里普克-普拉特克集理论</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Kripke-Platek_set_theory" class="extiw" title="en:Kripke-Platek set theory"><span lang="en" dir="auto">Kripke-Platek set theory</span></a></span>）</span></span>，省略了<a href="/wiki/%E6%97%A0%E7%A9%B7%E5%85%AC%E7%90%86" title="无穷公理">无穷公理</a>、<a href="/wiki/%E5%B9%82%E9%9B%86%E5%85%AC%E7%90%86" title="幂集公理">幂集公理</a>和<a href="/wiki/%E9%80%89%E6%8B%A9%E5%85%AC%E7%90%86" title="选择公理">选择公理</a>，削弱了<a href="/wiki/%E5%88%86%E7%B1%BB%E5%85%AC%E7%90%86" title="分类公理">分类公理</a>和<a href="/wiki/%E6%9B%BF%E4%BB%A3%E5%85%AC%E7%90%86" title="替代公理">替代公理</a>的公理架構。</li></ul></li> <li>由集合和<a href="/wiki/%E9%A1%9E_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="類 (數學)">真類</a>構成：這類系統包括<a href="/wiki/%E9%A6%AE%C2%B7%E8%AB%BE%E4%BC%8A%E6%9B%BC-%E5%8D%9A%E5%85%A7%E6%96%AF-%E5%93%A5%E5%BE%B7%E7%88%BE%E9%9B%86%E5%90%88%E8%AB%96" class="mw-redirect" title="馮·諾伊曼-博內斯-哥德爾集合論">馮·諾伊曼-博內斯-哥德爾集合論</a>，是設計生成同 ZFC同樣結果的集合論公理系統，但只有有限數目的公理而不使用<a href="/wiki/%E5%85%AC%E7%90%86%E6%A8%A1%E5%BC%8F" title="公理模式">公理模式</a>。單論只涉及集合的內容，此理論的強度和ZFC相當。另外比ZFC強的<span class="ilh-all" data-orig-title="Morse-Kelley集合論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Morse-Kelley set theory"><span class="ilh-page"><a href="/w/index.php?title=Morse-Kelley%E9%9B%86%E5%90%88%E8%AB%96&action=edit&redlink=1" class="new" title="Morse-Kelley集合論（页面不存在）">Morse-Kelley集合論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Morse-Kelley_set_theory" class="extiw" title="en:Morse-Kelley set theory"><span lang="en" dir="auto">Morse-Kelley set theory</span></a></span>）</span></span>及<span class="ilh-all" data-orig-title="Tarski–Grothendieck集合論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tarski–Grothendieck set theory"><span class="ilh-page"><a href="/w/index.php?title=Tarski%E2%80%93Grothendieck%E9%9B%86%E5%90%88%E8%AB%96&action=edit&redlink=1" class="new" title="Tarski–Grothendieck集合論（页面不存在）">Tarski–Grothendieck集合論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory" class="extiw" title="en:Tarski–Grothendieck set theory"><span lang="en" dir="auto">Tarski–Grothendieck set theory</span></a></span>）</span></span>也屬於這一類。</li></ul> <p>可以修改上述系統，允許<a href="/wiki/%E5%9F%BA%E6%9C%AC%E5%85%83%E7%B4%A0" title="基本元素">基本元素</a>（urelement）的存在，基本元素不是集合，因此本身也沒有成員，但基本元素可以是其他集合中的成員。 </p><p><a href="/wiki/%E6%96%B0%E5%9F%BA%E7%A1%80%E9%9B%86%E5%90%88%E8%AE%BA" title="新基础集合论">新基础集合论</a>的系統NFU（允許基本元素）及NF（不允許基本元素）不是以累计层次為基礎，NFU和NF有一個「包括所有物件的集合」，此外，每個集合都有其補集。這裡基本元素存在與否是個關鍵問題，因為NF給出了選擇公理的反例，而NFU則不會。新基礎集合論和<a href="/wiki/%E6%AD%A3%E9%9B%86%E5%90%88%E8%AE%BA" title="正集合论">正集合論</a>是已被提出的<a href="/wiki/%E5%8F%AF%E6%9B%BF%E4%BB%A3%E7%9A%84%E9%9B%86%E5%90%88%E8%AE%BA" title="可替代的集合论">可替代的集合論</a>之中的一部份。 </p><p><span class="ilh-all" data-orig-title="建構式集合論" data-lang-code="en" data-lang-name="英语" data-foreign-title="constructive set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%BB%BA%E6%A7%8B%E5%BC%8F%E9%9B%86%E5%90%88%E8%AB%96&action=edit&redlink=1" class="new" title="建構式集合論（页面不存在）">建構式集合論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/constructive_set_theory" class="extiw" title="en:constructive set theory"><span lang="en" dir="auto">constructive set theory</span></a></span>）</span></span>的系統，像是CST（建構式集合論）、CZF（建構式策梅洛-弗蘭克爾集合論）及IZF（直覺式策梅洛-弗蘭克爾集合論）等，將其集合公理以<a href="/wiki/%E7%9B%B4%E8%A7%89%E4%B8%BB%E4%B9%89%E9%80%BB%E8%BE%91" title="直觉主义逻辑">直觉主义逻辑</a>來表示，而不是使用<a href="/wiki/%E4%B8%80%E9%9A%8E%E9%82%8F%E8%BC%AF" class="mw-redirect" title="一階邏輯">一階邏輯</a>。其他的一些系統接受標準的一階邏輯，但是允許非標準的隸屬關係，包括<a href="/wiki/%E7%B2%97%E9%9B%86%E5%90%88" title="粗集合">粗集合</a>及<a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集</a>，其中表示隸屬關係的<a href="/wiki/%E5%8E%9F%E5%AD%90%E5%85%AC%E5%BC%8F" title="原子公式">原子公式</a>數值不只是單純的「真」或是「假」。ZFC中的<a href="/wiki/%E5%B8%83%E6%9E%97%E5%80%BC%E6%A8%A1%E5%9E%8B" class="mw-redirect" title="布林值模型">布林值模型</a>也是類似的概念。 </p><p><span class="ilh-all" data-orig-title="內集合論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Internal set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%A7%E9%9B%86%E5%90%88%E8%AB%96&action=edit&redlink=1" class="new" title="內集合論（页面不存在）">內集合論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Internal_set_theory" class="extiw" title="en:Internal set theory"><span lang="en" dir="auto">Internal set theory</span></a></span>）</span></span>是ZFC集合論的擴張，允許<a href="/wiki/%E7%84%A1%E7%AA%AE%E5%B0%8F%E9%87%8F" title="無窮小量">無窮小量</a>和其他「非標準」的數字存在，由<span class="ilh-all" data-orig-title="爱德华·尼尔森" data-lang-code="en" data-lang-name="英语" data-foreign-title="Edward Nelson"><span class="ilh-page"><a href="/w/index.php?title=%E7%88%B1%E5%BE%B7%E5%8D%8E%C2%B7%E5%B0%BC%E5%B0%94%E6%A3%AE&action=edit&redlink=1" class="new" title="爱德华·尼尔森（页面不存在）">爱德华·尼尔森</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Edward_Nelson" class="extiw" title="en:Edward Nelson"><span lang="en" dir="auto">Edward Nelson</span></a></span>）</span></span>在1977年提出。 </p> <div class="mw-heading mw-heading2"><h2 id="應用"><span id=".E6.87.89.E7.94.A8"></span>應用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=5" title="编辑章节：應用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>許多數學概念可以只用集合論的概念來準確定義。例如像<a href="/wiki/%E5%9B%BE" class="mw-disambig" title="图">圖</a>、<a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a>、<a href="/wiki/%E7%8E%AF_(%E4%BB%A3%E6%95%B0)" title="环 (代数)">環</a>和<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a>等數學結構都可以用滿足特定公理性質的集合來定義。在數學領域中，<a href="/wiki/%E7%AD%89%E4%BB%B7%E5%85%B3%E7%B3%BB" title="等价关系">等价关系</a>及<a href="/wiki/%E5%BA%8F%E7%90%86%E8%AE%BA" title="序理论">序关系</a>無所不在，而數學<a href="/wiki/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" title="二元关系">关系</a>的理論也可以用集合論來描述。 </p><p>對許多數學理論而言，集合論也是很有發展性的基礎系統。從集合論刊在《數學原理》的第一卷起，許多數學家聲稱大部份甚至全部的數學定理都可以用恰當設計的一些集合論公理來證明，其中可能會配合許多定義的加強，可能使用<a href="/wiki/%E4%B8%80%E9%9A%8E%E9%82%8F%E8%BC%AF" class="mw-redirect" title="一階邏輯">一階邏輯</a>或<a href="/wiki/%E4%BA%8C%E9%9A%8E%E9%82%8F%E8%BC%AF" title="二階邏輯">二階邏輯</a>。例如有關<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B8" class="mw-redirect" title="自然數">自然數</a>或是<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>的性質就可以用集合論來推導，每個數系都等同為某個由<a href="/wiki/%E7%AD%89%E5%83%B9%E9%A1%9E" class="mw-redirect" title="等價類">等價類</a>組成的集合（從在某個<a href="/wiki/%E7%84%A1%E9%99%90%E9%9B%86" class="mw-redirect" title="無限集">無限集</a>上的<a href="/wiki/%E7%AD%89%E4%BB%B7%E5%85%B3%E7%B3%BB" title="等价关系">等价关系</a>所得）。 </p><p>集合論在<a href="/wiki/%E6%95%B8%E5%AD%B8%E5%88%86%E6%9E%90" class="mw-redirect" title="數學分析">數學分析</a>、<a href="/wiki/%E6%8B%93%E6%92%B2%E5%AD%B8" class="mw-redirect" title="拓撲學">拓撲學</a>、<a href="/wiki/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0" title="抽象代数">抽象代数</a>及<a href="/wiki/%E9%9B%A2%E6%95%A3%E6%95%B8%E5%AD%B8" class="mw-redirect" title="離散數學">離散數學</a>中的基礎地位比較沒有爭議，數學家接受這些領域中的定理（或是比較基礎的定理）可以由集合論中的公理及適當的定義推導出來。因為用集合論證明複雜理論的推導過程比一般的推導過程長很多，只有少量這種的證明被<a href="/wiki/%E5%AE%9A%E7%90%86%E6%9C%BA%E5%99%A8%E8%AF%81%E6%98%8E" class="mw-redirect" title="定理机器证明">正式驗證</a>過。一個稱為<a href="/wiki/Metamath" title="Metamath">Metamath</a>的驗證計劃，以ZFC集合论為起點，使用<a href="/wiki/%E4%B8%80%E9%9A%8E%E9%82%8F%E8%BC%AF" class="mw-redirect" title="一階邏輯">一階邏輯</a>來進行證明，包括了超過一萬個定理證明的推導。 </p> <div class="mw-heading mw-heading2"><h2 id="研究領域"><span id=".E7.A0.94.E7.A9.B6.E9.A0.98.E5.9F.9F"></span>研究領域</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=6" title="编辑章节：研究領域"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>集合論是數學的主要研究領域之一，其中也有許多和其他領域相關的子領域。 </p> <div class="mw-heading mw-heading3"><h3 id="組合集合論"><span id=".E7.B5.84.E5.90.88.E9.9B.86.E5.90.88.E8.AB.96"></span>組合集合論</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=7" title="编辑章节：組合集合論"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>組合集合論也稱為<span class="ilh-all" data-orig-title="無限組合數學" data-lang-code="en" data-lang-name="英语" data-foreign-title="Infinitary combinatorics"><span class="ilh-page"><a href="/w/index.php?title=%E7%84%A1%E9%99%90%E7%B5%84%E5%90%88%E6%95%B8%E5%AD%B8&action=edit&redlink=1" class="new" title="無限組合數學（页面不存在）">無限組合數學</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Infinitary_combinatorics" class="extiw" title="en:Infinitary combinatorics"><span lang="en" dir="auto">Infinitary combinatorics</span></a></span>）</span></span>，將有限的<a href="/wiki/%E7%B5%84%E5%90%88%E6%95%B8%E5%AD%B8" class="mw-redirect" title="組合數學">組合數學</a>延伸到<a href="/wiki/%E7%84%A1%E9%99%90%E9%9B%86" class="mw-redirect" title="無限集">無限集</a>中。組合集合論包括<a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)#基數算術" title="基数 (数学)">基數算術</a>的研究，以及<a href="/wiki/%E6%8B%89%E5%A7%86%E9%BD%90%E5%AE%9A%E7%90%86" title="拉姆齐定理">拉姆齐定理</a>的擴展，例如<span class="ilh-all" data-orig-title="艾狄胥–拉多定理" data-lang-code="en" data-lang-name="英语" data-foreign-title="Erdős–Rado theorem"><span class="ilh-page"><a href="/w/index.php?title=%E8%89%BE%E7%8B%84%E8%83%A5%E2%80%93%E6%8B%89%E5%A4%9A%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="艾狄胥–拉多定理（页面不存在）">艾狄胥–拉多定理</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Rado_theorem" class="extiw" title="en:Erdős–Rado theorem"><span lang="en" dir="auto">Erdős–Rado theorem</span></a></span>）</span></span>。 </p> <div class="mw-heading mw-heading3"><h3 id="描述集合論"><span id=".E6.8F.8F.E8.BF.B0.E9.9B.86.E5.90.88.E8.AB.96"></span>描述集合論</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=8" title="编辑章节：描述集合論"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%8F%8F%E8%BF%B0%E9%9B%86%E5%90%88%E8%AB%96" title="描述集合論">描述集合論</a></div> <p>描述集合論是關於<a href="/wiki/%E5%AE%9E%E7%9B%B4%E7%BA%BF" class="mw-disambig" title="实直线">實直線</a>或<a href="/wiki/%E6%B3%A2%E8%98%AD%E7%A9%BA%E9%96%93" class="mw-redirect" title="波蘭空間">波蘭空間</a>上子集的研究。描述集合論是從對<span class="ilh-all" data-orig-title="波莱尔层次" data-lang-code="en" data-lang-name="英语" data-foreign-title="Borel hierarchy"><span class="ilh-page"><a href="/w/index.php?title=%E6%B3%A2%E8%8E%B1%E5%B0%94%E5%B1%82%E6%AC%A1&action=edit&redlink=1" class="new" title="波莱尔层次（页面不存在）">波莱尔层次</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Borel_hierarchy" class="extiw" title="en:Borel hierarchy"><span lang="en" dir="auto">Borel hierarchy</span></a></span>）</span></span>中<a href="/wiki/%E7%82%B9%E9%9B%86" class="mw-redirect mw-disambig" title="点集">点集</a>研究開始，後來延伸到更複雜的層次，像是<span class="ilh-all" data-orig-title="射影层次" data-lang-code="en" data-lang-name="英语" data-foreign-title="projective hierarchy"><span class="ilh-page"><a href="/w/index.php?title=%E5%B0%84%E5%BD%B1%E5%B1%82%E6%AC%A1&action=edit&redlink=1" class="new" title="射影层次（页面不存在）">射影层次</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/projective_hierarchy" class="extiw" title="en:projective hierarchy"><span lang="en" dir="auto">projective hierarchy</span></a></span>）</span></span>及<span class="ilh-all" data-orig-title="魏吉层次" data-lang-code="en" data-lang-name="英语" data-foreign-title="Wadge hierarchy"><span class="ilh-page"><a href="/w/index.php?title=%E9%AD%8F%E5%90%89%E5%B1%82%E6%AC%A1&action=edit&redlink=1" class="new" title="魏吉层次（页面不存在）">魏吉层次</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Wadge_hierarchy" class="extiw" title="en:Wadge hierarchy"><span lang="en" dir="auto">Wadge hierarchy</span></a></span>）</span></span>。<a href="/wiki/%E6%B3%A2%E8%8E%B1%E5%B0%94%E9%9B%86" class="mw-redirect" title="波莱尔集">波莱尔集</a>中的許多性質可以建立在包括選擇公理的策梅洛-弗蘭克爾集合論上，但要證明更複雜的集合也符合這些性質的話，就需要有其他和<a href="/wiki/%E6%B1%BA%E5%AE%9A%E5%85%AC%E7%90%86" title="決定公理">決定公理</a>和<a href="/wiki/%E5%A4%A7%E5%9F%BA%E6%95%B0" title="大基数">大基数</a>有關的公理。 </p><p><span class="ilh-all" data-orig-title="有效描述集合論" data-lang-code="en" data-lang-name="英语" data-foreign-title="effective descriptive set theory"><span class="ilh-page"><a href="/w/index.php?title=%E6%9C%89%E6%95%88%E6%8F%8F%E8%BF%B0%E9%9B%86%E5%90%88%E8%AB%96&action=edit&redlink=1" class="new" title="有效描述集合論（页面不存在）">有效描述集合論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/effective_descriptive_set_theory" class="extiw" title="en:effective descriptive set theory"><span lang="en" dir="auto">effective descriptive set theory</span></a></span>）</span></span>介於集合論和<a href="/wiki/%E9%80%92%E5%BD%92%E8%AE%BA" title="递归论">递归论</a>之間，包括對<span class="ilh-all" data-orig-title="淺體點集" data-lang-code="en" data-lang-name="英语" data-foreign-title="lightface pointclass"><span class="ilh-page"><a href="/w/index.php?title=%E6%B7%BA%E9%AB%94%E9%BB%9E%E9%9B%86&action=edit&redlink=1" class="new" title="淺體點集（页面不存在）">淺體點集</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/lightface_pointclass" class="extiw" title="en:lightface pointclass"><span lang="en" dir="auto">lightface pointclass</span></a></span>）</span></span>的研究，和<span class="ilh-all" data-orig-title="超算術理論" data-lang-code="en" data-lang-name="英语" data-foreign-title="hyperarithmetical theory"><span class="ilh-page"><a href="/w/index.php?title=%E8%B6%85%E7%AE%97%E8%A1%93%E7%90%86%E8%AB%96&action=edit&redlink=1" class="new" title="超算術理論（页面不存在）">超算術理論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/hyperarithmetical_theory" class="extiw" title="en:hyperarithmetical theory"><span lang="en" dir="auto">hyperarithmetical theory</span></a></span>）</span></span>緊密相關。許多情形下，描述集合論的結果也可以用有效描述集合論來表示。有時，會先利用有效描述集合論來證明，再將其延伸（相對化），使其應用範圍更廣。 </p><p>描述集合論的當代研究包括<span class="ilh-all" data-orig-title="波萊爾等價關係" data-lang-code="en" data-lang-name="英语" data-foreign-title="Borel equivalence relation"><span class="ilh-page"><a href="/w/index.php?title=%E6%B3%A2%E8%90%8A%E7%88%BE%E7%AD%89%E5%83%B9%E9%97%9C%E4%BF%82&action=edit&redlink=1" class="new" title="波萊爾等價關係（页面不存在）">波萊爾等價關係</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Borel_equivalence_relation" class="extiw" title="en:Borel equivalence relation"><span lang="en" dir="auto">Borel equivalence relation</span></a></span>）</span></span>及一些更複雜的可定義<a href="/wiki/%E7%AD%89%E5%83%B9%E9%97%9C%E4%BF%82" class="mw-redirect" title="等價關係">等價關係</a>。描述集合論在許多數學領域的<a href="/wiki/%E4%B8%8D%E5%8F%98%E9%87%8F_(%E6%95%B0%E5%AD%A6)" class="mw-redirect" title="不变量 (数学)">不變量</a>研究都很重要。 </p> <div class="mw-heading mw-heading3"><h3 id="大基數"><span id=".E5.A4.A7.E5.9F.BA.E6.95.B8"></span>大基數</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=9" title="编辑章节：大基數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%A4%A7%E5%9F%BA%E6%95%B0" title="大基数">大基數</a></div> <p><b>大基數</b>是具有特殊性質的基數。許多不同的性質也有研究，例如<a href="/wiki/%E4%B8%8D%E5%8F%AF%E9%81%94%E5%9F%BA%E6%95%B8" title="不可達基數">不可達基數</a>和<a href="/wiki/%E5%8F%AF%E6%B8%AC%E5%9F%BA%E6%95%B8" title="可測基數">可測基數</a>。此類性質通常推出該基數必定非常大，且其存在性不能在<a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅洛-弗蘭克爾集合論</a>證明。 </p> <div class="mw-heading mw-heading3"><h3 id="模糊集"><span id=".E6.A8.A1.E7.B3.8A.E9.9B.86"></span>模糊集</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=10" title="编辑章节：模糊集"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集</a></div> <p>在康托尔定義的樸素集合論及後來發展的ZFC集合論中，一個物件和一個集合的關係只有二種：是成員或者不是成員。<a href="/wiki/%E7%9B%A7%E8%8F%B2%E7%89%B9%C2%B7%E6%BE%A4%E5%BE%B7" class="mw-redirect" title="盧菲特·澤德">盧菲特·澤德</a>在<a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集</a>中放寬上述的限制，物件有對於集合的歸屬度（degree of membership），是一個介於0到1之間的數字。例如有關一個人對於「身材高大的人」集合的歸屬度不是簡單的是或不是，而是一個數值，例如0.75。 </p> <div class="mw-heading mw-heading3"><h3 id="力迫"><span id=".E5.8A.9B.E8.BF.AB"></span>力迫</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=11" title="编辑章节：力迫"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%8A%9B%E8%BF%AB" title="力迫">力迫</a></div> <p><a href="/wiki/%E4%BF%9D%E7%BD%97%C2%B7%E5%AF%87%E6%81%A9" title="保罗·寇恩">保羅·寇恩</a>的一些工作是尋找一個ZFC的模型，使得<a href="/wiki/%E9%81%B8%E6%93%87%E5%85%AC%E7%90%86" class="mw-redirect" title="選擇公理">選擇公理</a>或<a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BB%9F%E5%81%87%E8%AE%BE" title="连续统假设">连续统假设</a>失效，在這過程中他發明了<a href="/wiki/%E5%8A%9B%E8%BF%AB" title="力迫">力迫</a>。力迫法是一種擴張模型的方法，在集合論的某模型中加入一些額外的集合，來產生一個較大的模型，這樣的模型會具有預期的性質（依賴於具體構造方式和原模型）。例如，在保罗·寇恩的構造中，他給原模型附加了額外的<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B8" class="mw-redirect" title="自然數">自然數</a>子集，而沒有更改原模型的任何<a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="基数 (数学)">基數</a>。力迫也是利用有窮方法（finitistic method）證明相對<a href="/wiki/%E4%B8%80%E8%87%B4%E6%80%A7_(%E9%82%8F%E8%BC%AF)" title="一致性 (邏輯)">一致性</a>的二種方法中的一種，另一個方法是<a href="/wiki/%E5%B8%83%E5%B0%94%E5%80%BC%E6%A8%A1%E5%9E%8B" title="布尔值模型">布尔值模型</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="對集合論的異議"><span id=".E5.B0.8D.E9.9B.86.E5.90.88.E8.AB.96.E7.9A.84.E7.95.B0.E8.AD.B0"></span>對集合論的異議</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=12" title="编辑章节：對集合論的異議"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一開始，有些數學家<span class="ilh-all" data-orig-title="康托爾理論的爭論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Controversy over Cantor's theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%BA%B7%E6%89%98%E7%88%BE%E7%90%86%E8%AB%96%E7%9A%84%E7%88%AD%E8%AB%96&action=edit&redlink=1" class="new" title="康托爾理論的爭論（页面不存在）">反對</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory" class="extiw" title="en:Controversy over Cantor's theory"><span lang="en" dir="auto">Controversy over Cantor's theory</span></a></span>）</span></span>將集合論當做<a href="/wiki/%E6%95%B8%E5%AD%B8%E5%9F%BA%E7%A4%8E" class="mw-redirect" title="數學基礎">數學基礎</a>，認為這只是一場含有「奇幻元素」的遊戲。對集合論最常見的反對意見來自<a href="/wiki/%E6%95%B0%E5%AD%A6%E7%BB%93%E6%9E%84%E4%B8%BB%E4%B9%89" class="mw-redirect" title="数学结构主义">數學結構主義者</a>（像是<a href="/wiki/%E5%88%A9%E5%A5%A5%E6%B3%A2%E5%BE%B7%C2%B7%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B" title="利奥波德·克罗内克">利奥波德·克罗内克</a>），他們認為數學多少都和計算有些關係的，但<a href="/wiki/%E6%9C%B4%E7%B4%A0%E9%9B%86%E5%90%88%E8%AE%BA" title="朴素集合论">樸素集合論</a>卻加入了非計算性的元素。 </p><p><span class="ilh-all" data-orig-title="埃里特·比修普" data-lang-code="en" data-lang-name="英语" data-foreign-title="Errett Bishop"><span class="ilh-page"><a href="/w/index.php?title=%E5%9F%83%E9%87%8C%E7%89%B9%C2%B7%E6%AF%94%E4%BF%AE%E6%99%AE&action=edit&redlink=1" class="new" title="埃里特·比修普（页面不存在）">埃里特·比修普</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Errett_Bishop" class="extiw" title="en:Errett Bishop"><span lang="en" dir="auto">Errett Bishop</span></a></span>）</span></span>駁斥集合論是「<a href="/wiki/%E4%B8%8A%E5%B8%9D" title="上帝">上帝</a>的數學，應該留給上帝」。而且，<a href="/wiki/%E8%B7%AF%E5%BE%B7%E7%BB%B4%E5%B8%8C%C2%B7%E7%BB%B4%E7%89%B9%E6%A0%B9%E6%96%AF%E5%9D%A6" title="路德维希·维特根斯坦">路德維希·維根斯坦</a>特別對無限的操作有疑問，這也和<a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅羅-弗蘭克爾集合論</a>有關。維根斯坦對於數學基礎的觀點曾被<span class="ilh-all" data-orig-title="保羅·貝奈斯" data-lang-code="en" data-lang-name="英语" data-foreign-title="Paul Bernays"><span class="ilh-page"><a href="/w/index.php?title=%E4%BF%9D%E7%BE%85%C2%B7%E8%B2%9D%E5%A5%88%E6%96%AF&action=edit&redlink=1" class="new" title="保羅·貝奈斯（页面不存在）">保羅·貝奈斯</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Paul_Bernays" class="extiw" title="en:Paul Bernays"><span lang="en" dir="auto">Paul Bernays</span></a></span>）</span></span>所批評，且被<span class="ilh-all" data-orig-title="克里斯平·賴特" data-lang-code="en" data-lang-name="英语" data-foreign-title="Crispin Wright"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%8B%E9%87%8C%E6%96%AF%E5%B9%B3%C2%B7%E8%B3%B4%E7%89%B9&action=edit&redlink=1" class="new" title="克里斯平·賴特（页面不存在）">克里斯平·賴特</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Crispin_Wright" class="extiw" title="en:Crispin Wright"><span lang="en" dir="auto">Crispin Wright</span></a></span>）</span></span>等人密切研究過<sup id="cite_ref-Philosophy_4-0" class="reference"><a href="#cite_note-Philosophy-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>。 </p><p><a href="/wiki/%E6%8B%93%E6%92%B2%E6%96%AF" title="拓撲斯">拓撲斯</a>理論曾被認為是傳統公理化集合論的另一種選擇。拓樸斯理論可以被用來解釋該集合論的各種替代方案，如<a href="/wiki/%E6%95%B0%E5%AD%A6%E7%BB%93%E6%9E%84%E4%B8%BB%E4%B9%89" class="mw-redirect" title="数学结构主义">數學結構主義</a>、<a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集合論</a>、有限集合論和<a href="/wiki/%E5%9B%BE%E7%81%B5%E6%9C%BA" title="图灵机">可計算</a>集合論等<sup id="cite_ref-Ferro_5-0" class="reference"><a href="#cite_note-Ferro-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="相關條目"><span id=".E7.9B.B8.E9.97.9C.E6.A2.9D.E7.9B.AE"></span>相關條目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=13" title="编辑章节：相關條目"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="navigation" aria-label="Portals" class="noprint portal plainlist tright" style="margin:0.5em 0 0.5em 1em;border:solid #aaa 1px"> <ul style="display:table;box-sizing:border-box;padding:0.1em;max-width:175px;background:var(--background-color-base,#f9f9f9);font-size:85%;line-height:110%;font-weight:bold"> <li 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title="Portal:数学">数学主题</a></span></li></ul></div> <style data-mw-deduplicate="TemplateStyles:r82655521">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox" style="font-size:small;"><style data-mw-deduplicate="TemplateStyles:r82655520">.mw-parser-output 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href="https://zh.wikiversity.org/wiki/Topic:%E9%9B%86%E5%90%88%E8%AB%96" class="extiw" title="v:Topic:集合論">集合论</a></b></div></div> </div> <ul><li><span class="ilh-all" data-orig-title="集合論主題列表" data-lang-code="en" data-lang-name="英语" data-foreign-title="List of set theory topics"><span class="ilh-page"><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AB%96%E4%B8%BB%E9%A1%8C%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="集合論主題列表（页面不存在）">集合論主題列表</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/List_of_set_theory_topics" class="extiw" title="en:List of set theory topics"><span lang="en" dir="auto">List of set theory topics</span></a></span>）</span></span></li> <li><a href="/wiki/%E7%B4%84%E7%95%A5%E9%9B%86%E5%90%88" class="mw-redirect" title="約略集合">約略集合</a>論提供了一個以上下近似來表示集合的方法。</li> <li><span class="ilh-all" data-orig-title="音樂集合理論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Set theory (music)"><span class="ilh-page"><a href="/w/index.php?title=%E9%9F%B3%E6%A8%82%E9%9B%86%E5%90%88%E7%90%86%E8%AB%96&action=edit&redlink=1" class="new" title="音樂集合理論（页面不存在）">音樂集合理論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Set_theory_(music)" class="extiw" title="en:Set theory (music)"><span lang="en" dir="auto">Set theory (music)</span></a></span>）</span></span>將<a href="/wiki/%E7%B5%84%E5%90%88%E6%95%B8%E5%AD%B8" class="mw-redirect" title="組合數學">組合數學</a>和<a href="/wiki/%E7%BE%A4%E8%AB%96" class="mw-redirect" title="群論">群論</a>應用在音樂上；但除了使用<a href="/wiki/%E6%9C%89%E9%99%90%E9%9B%86" class="mw-redirect" title="有限集">有限集</a>外，事實上它和數學中任何一種類型的集合論都沒多大關係。最近兩個年代以來，音樂中的<span class="ilh-all" data-orig-title="轉換理論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Transformational theory"><span class="ilh-page"><a href="/w/index.php?title=%E8%BD%89%E6%8F%9B%E7%90%86%E8%AB%96&action=edit&redlink=1" class="new" title="轉換理論（页面不存在）">轉換理論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Transformational_theory" class="extiw" title="en:Transformational theory"><span lang="en" dir="auto">Transformational theory</span></a></span>）</span></span>已較嚴謹地採用數學集合論中的概念。</li> <li><a href="/wiki/%E7%AF%84%E7%96%87%E8%AB%96" class="mw-redirect" title="範疇論">範疇論</a>也以抽象的方法來處理數學概念。</li> <li><a href="/wiki/%E9%97%9C%E8%81%AF%E6%A8%A1%E5%9E%8B" class="mw-redirect" title="關聯模型">關聯模型</a>有借用集合论中的一些概念。</li></ul> <div class="mw-heading mw-heading2"><h2 id="參考資料"><span id=".E5.8F.83.E8.80.83.E8.B3.87.E6.96.99"></span>參考資料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=14" title="编辑章节：參考資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation journal">Milinowski, A.; Cantor, G. <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583">Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen.</a>. Journal für die reine und angewandte Mathematik. 1874, <b>77</b>: 258–262 <span class="reference-accessdate"> [<span class="nowrap">2013-05-17</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://archive.today/20120604145721/http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583">存档</a>于2012-06-04） <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到德语网页">（德语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E9%9B%86%E5%90%88%E8%AE%BA&rft.atitle=Ueber+eine+Eigenschaft+des+Inbegriffs+aller+reellen+algebraischen+Zahlen.&rft.au=Cantor%2C+G.&rft.aufirst=A.&rft.aulast=Milinowski&rft.date=1874&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.pages=258-262&rft.volume=77&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fmain%2Fdms%2Fimg%2F%3FPPN%3DGDZPPN002155583&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFJohnson1972" class="citation">Johnson, Philip, A History of Set Theory, Prindle, Weber & Schmidt, 1972, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-87150-154-6" title="Special:网络书源/0-87150-154-6"><span title="国际标准书号">ISBN</span> 0-87150-154-6</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E9%9B%86%E5%90%88%E8%AE%BA&rft.aufirst=Philip&rft.aulast=Johnson&rft.btitle=A+History+of+Set+Theory&rft.date=1972&rft.genre=book&rft.isbn=0-87150-154-6&rft.pub=Prindle%2C+Weber+%26+Schmidt&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFBolzano1975" class="citation"><a href="/wiki/Bernard_Bolzano" class="mw-redirect" title="Bernard Bolzano">Bolzano, Bernard</a>, Berg, Jan , 编, Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., Vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag: 152, 1975, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/3-7728-0466-7" title="Special:网络书源/3-7728-0466-7"><span title="国际标准书号">ISBN</span> 3-7728-0466-7</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E9%9B%86%E5%90%88%E8%AE%BA&rft.aufirst=Bernard&rft.aulast=Bolzano&rft.btitle=Einleitung+zur+Gr%C3%B6%C3%9Fenlehre+und+erste+Begriffe+der+allgemeinen+Gr%C3%B6%C3%9Fenlehre&rft.date=1975&rft.genre=book&rft.isbn=3-7728-0466-7&rft.pages=152&rft.place=Stuttgart%2C+Bad+Cannstatt&rft.pub=Friedrich+Frommann+Verlag&rft.series=Bernard-Bolzano-Gesamtausgabe%2C+edited+by+Eduard+Winter+et+al.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Philosophy-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Philosophy_4-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Francis, John. <a rel="nofollow" class="external text" href="https://books.google.com.tw/books?id=DuyMjOwWWnUC&pg=PA86">Philosophy Of Mathematics</a>. Global Vision Publishing House. 2008-08: 86. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-81-8220-267-2" title="Special:网络书源/978-81-8220-267-2"><span title="国际标准书号">ISBN</span> 978-81-8220-267-2</a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E9%9B%86%E5%90%88%E8%AE%BA&rft.aufirst=John&rft.aulast=Francis&rft.btitle=Philosophy+Of+Mathematics&rft.date=2008-08&rft.genre=book&rft.isbn=978-81-8220-267-2&rft.pages=86&rft.pub=Global+Vision+Publishing+House&rft_id=https%3A%2F%2Fbooks.google.com.tw%2Fbooks%3Fid%3DDuyMjOwWWnUC%26pg%3DPA86&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Ferro-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ferro_5-0">^</a></b></span> <span class="reference-text"><cite class="citation journal">Ferro, Alfredo; Omodeo, Eugenio G.; Schwartz, Jacob T. <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160330503">Decision procedures for elementary sublanguages of set theory. I. Multi-level syllogistic and some extensions</a>. Communications on Pure and Applied Mathematics. 1980-09, <b>33</b> (5): 599–608 <span class="reference-accessdate"> [<span class="nowrap">2022-03-23</span>]</span>. <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160330503"><span title="數位物件識別號">doi:10.1002/cpa.3160330503</span></a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20220118061426/https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160330503">存档</a>于2022-01-18） <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E9%9B%86%E5%90%88%E8%AE%BA&rft.atitle=Decision+procedures+for+elementary+sublanguages+of+set+theory.+I.+Multi-level+syllogistic+and+some+extensions&rft.au=Omodeo%2C+Eugenio+G.&rft.au=Schwartz%2C+Jacob+T.&rft.aufirst=Alfredo&rft.aulast=Ferro&rft.date=1980-09&rft.genre=article&rft.issue=5&rft.jtitle=Communications+on+Pure+and+Applied+Mathematics&rft.pages=599-608&rft.volume=33&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fcpa.3160330503&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160330503&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="額外閱讀"><span id=".E9.A1.8D.E5.A4.96.E9.96.B1.E8.AE.80"></span>額外閱讀</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=15" title="编辑章节：額外閱讀"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFHerbert_B._Enderton1977" class="citation">Herbert B. Enderton, Elements of Set Theory 1nd, Academic Press, 1977, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0122384400" title="Special:网络书源/978-0122384400"><span title="国际标准书号">ISBN</span> 978-0122384400</a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E9%9B%86%E5%90%88%E8%AE%BA&rft.au=Herbert+B.+Enderton&rft.btitle=Elements+of+Set+Theory&rft.date=1977&rft.edition=1nd&rft.genre=book&rft.isbn=978-0122384400&rft.pub=Academic+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>,集合論入門書。</li></ul> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=16" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%8F%B2%E4%B8%B9%E4%BD%9B%E5%93%B2%E5%AD%B8%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="史丹佛哲學百科全書">史丹佛哲學百科全書</a>相關條目: </p> <ul><li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/set-theory/">集合論</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210413165506/https://plato.stanford.edu/entries/set-theory/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）<span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span></li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/settheory-early/">集合論的早期發展</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210512135148/https://plato.stanford.edu/entries/settheory-early/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）<span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span></li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/zermelo-set-theory/">策梅洛的公理化集合論</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210506151817/https://plato.stanford.edu/entries/zermelo-set-theory/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）<span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span></li></ul> <p><a rel="nofollow" class="external text" href="https://www.iep.utm.edu/set-theo/">集合論</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210118112958/https://www.iep.utm.edu/set-theo/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）<span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span>，網路哲學百科全書 </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.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 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"></div><div role="navigation" class="navbox" aria-labelledby="集合论" style="padding:3px"><table class="nowraplinks mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar 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href="/wiki/Template:%E9%9B%86%E5%90%88%E8%AE%BA" title="Template:集合论"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E9%9B%86%E5%90%88%E8%AE%BA" title="Template talk:集合论"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E9%9B%86%E5%90%88%E8%AE%BA" title="Special:编辑页面/Template:集合论"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="集合论" style="font-size:110%;margin:0 5em"><a class="mw-selflink selflink">集合论</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%80%89%E6%8B%A9%E5%85%AC%E7%90%86" title="选择公理">选择</a> <ul><li><a href="/wiki/%E5%8F%AF%E6%95%B0%E9%80%89%E6%8B%A9%E5%85%AC%E7%90%86" title="可数选择公理">可数</a></li> <li><a href="/wiki/%E4%BE%9D%E8%B3%B4%E9%81%B8%E6%93%87%E5%85%AC%E7%90%86" title="依賴選擇公理">依賴</a></li></ul></li> <li><a href="/wiki/%E5%A4%96%E5%BB%B6%E5%85%AC%E7%90%86" title="外延公理">外延</a></li> <li><a href="/wiki/%E6%97%A0%E7%A9%B7%E5%85%AC%E7%90%86" title="无穷公理">无穷</a></li> <li><a href="/wiki/%E9%85%8D%E5%AF%B9%E5%85%AC%E7%90%86" title="配对公理">配对</a></li> <li><a href="/wiki/%E5%B9%82%E9%9B%86%E5%85%AC%E7%90%86" title="幂集公理">幂集</a></li> <li><a href="/wiki/%E6%AD%A3%E5%88%99%E6%80%A7%E5%85%AC%E7%90%86" title="正则性公理">正则性</a></li> <li><a href="/wiki/%E5%B9%B6%E9%9B%86%E5%85%AC%E7%90%86" title="并集公理">并集</a></li> <li><a href="/wiki/%E9%A6%AC%E4%B8%81%E5%85%AC%E7%90%86" title="馬丁公理">马丁公理</a></li></ul> <ul><li><a href="/wiki/%E5%85%AC%E7%90%86%E6%A8%A1%E5%BC%8F" title="公理模式">公理模式</a> <ul><li><a href="/wiki/%E6%9B%BF%E4%BB%A3%E5%85%AC%E7%90%86" title="替代公理">替代</a></li> <li><a href="/wiki/%E5%88%86%E7%B1%BB%E5%85%AC%E7%90%86" title="分类公理">分类</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/%E7%BB%B4%E6%81%A9%E5%9B%BE" title="维恩图"><img alt="Venn diagram of set intersection" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/100px-Venn_A_intersect_B.svg.png" decoding="async" width="100" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/150px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/200px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">运算</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%84%BF%E7%A7%AF" title="笛卡儿积">笛卡儿积</a></li> <li><a href="/wiki/%E5%BE%B7%E6%91%A9%E6%A0%B9%E5%AE%9A%E5%BE%8B" title="德摩根定律">德摩根定律</a></li> <li><a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a></li> <li><a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">冪集</a></li> <li><a href="/wiki/%E8%A1%A5%E9%9B%86" title="补集">补集</a></li> <li><a href="/wiki/%E5%AF%B9%E7%A7%B0%E5%B7%AE" title="对称差">对称差</a></li> <li><a href="/wiki/%E5%B9%B6%E9%9B%86" title="并集">并集</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>概念</li><li>方法</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8A%BF_(%E6%95%B0%E5%AD%A6)" title="势 (数学)">势</a></li> <li><a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="基数 (数学)">基数</a>（<a href="/wiki/%E5%A4%A7%E5%9F%BA%E6%95%B0" title="大基数">大基数</a>）</li> <li><a href="/wiki/%E7%B1%BB_(%E6%95%B0%E5%AD%A6)" title="类 (数学)">类</a></li> <li><span class="ilh-all" data-orig-title="可构造全集" data-lang-code="en" data-lang-name="英语" data-foreign-title="Constructible universe"><span class="ilh-page"><a href="/w/index.php?title=%E5%8F%AF%E6%9E%84%E9%80%A0%E5%85%A8%E9%9B%86&action=edit&redlink=1" class="new" title="可构造全集（页面不存在）">可构造全集</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Constructible_universe" class="extiw" title="en:Constructible universe"><span lang="en" dir="auto">Constructible universe</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BB%9F%E5%81%87%E8%AE%BE" title="连续统假设">连续统假设</a></li> <li><a href="/wiki/%E5%B0%8D%E8%A7%92%E8%AB%96%E8%AD%89%E6%B3%95" title="對角論證法">對角論證法</a></li> <li><a href="/wiki/%E5%85%83%E7%B4%A0_(%E6%95%B8%E5%AD%B8)" title="元素 (數學)">元素</a> <ul><li><a href="/wiki/%E6%9C%89%E5%BA%8F%E5%AF%B9" title="有序对">有序对</a></li> <li><a href="/wiki/%E5%85%83%E7%BB%84" title="元组">元组</a></li></ul></li> <li><a href="/wiki/%E9%9B%86%E5%90%88%E6%97%8F" title="集合族">集合族</a></li> <li><a href="/wiki/%E5%8A%9B%E8%BF%AB" title="力迫">力迫</a></li> <li><a href="/wiki/%E5%8F%8C%E5%B0%84" title="双射">一一对应</a></li> <li><a href="/wiki/%E5%BA%8F%E6%95%B0" title="序数">序数</a></li> <li><a href="/wiki/%E8%B6%85%E9%99%90%E5%BD%92%E7%BA%B3%E6%B3%95" title="超限归纳法">超限归纳法</a></li> <li><a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏图</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>类型</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8F%AF%E6%95%B8%E9%9B%86" title="可數集">可數集</a></li> <li><a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a></li> <li><a href="/wiki/%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="有限集合">有限集合</a>（<a href="/wiki/%E7%BB%A7%E6%89%BF%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="继承有限集合">继承有限集合</a>）</li> <li><a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集</a></li> <li><a href="/wiki/%E6%97%A0%E9%99%90%E9%9B%86%E5%90%88" title="无限集合">无限集合</a></li> <li><a href="/wiki/%E9%80%92%E5%BD%92%E9%9B%86%E5%90%88" title="递归集合">递归集合</a></li> <li><a href="/wiki/%E5%AD%90%E9%9B%86" title="子集">子集</a></li> <li><a href="/wiki/%E4%BC%A0%E9%80%92%E9%9B%86%E5%90%88" title="传递集合">传递集合</a></li> <li><a href="/wiki/%E4%B8%8D%E5%8F%AF%E6%95%B8%E9%9B%86" title="不可數集">不可數集</a></li> <li><span class="ilh-all" data-orig-title="泛集" data-lang-code="en" data-lang-name="英语" data-foreign-title="Universal set"><span class="ilh-page"><a href="/w/index.php?title=%E6%B3%9B%E9%9B%86&action=edit&redlink=1" class="new" title="泛集（页面不存在）">泛集</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Universal_set" class="extiw" title="en:Universal set"><span lang="en" dir="auto">Universal set</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">理论</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8F%AF%E6%9B%BF%E4%BB%A3%E7%9A%84%E9%9B%86%E5%90%88%E8%AE%BA" title="可替代的集合论">可替代的集合论</a></li> <li><a class="mw-selflink selflink">集合论</a></li> <li><a href="/wiki/%E6%9C%B4%E7%B4%A0%E9%9B%86%E5%90%88%E8%AE%BA" title="朴素集合论">朴素集合论</a></li> <li><a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94%E5%AE%9A%E7%90%86" title="康托尔定理">康托尔定理</a></li></ul> <ul><li><a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛集合论">策梅洛</a> <ul><li><span class="ilh-all" data-orig-title="广义集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="General set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%B9%BF%E4%B9%89%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="广义集合论（页面不存在）">广义</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/General_set_theory" class="extiw" title="en:General set theory"><span lang="en" dir="auto">General set theory</span></a></span>）</span></span></li></ul></li> <li>《<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%8E%9F%E7%90%86" title="数学原理">数学原理</a>》 <ul><li><a href="/wiki/%E6%96%B0%E5%9F%BA%E7%A1%80%E9%9B%86%E5%90%88%E8%AE%BA" title="新基础集合论">新基础</a></li></ul></li> <li><a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅洛-弗兰克</a> <ul><li><a href="/wiki/%E5%86%AF%E8%AF%BA%E4%BC%8A%E6%9B%BC-%E5%8D%9A%E5%86%85%E6%96%AF-%E5%93%A5%E5%BE%B7%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="冯诺伊曼-博内斯-哥德尔集合论">冯诺伊曼-博内斯-哥德尔</a> <ul><li><span class="ilh-all" data-orig-title="Morse–Kelley集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Morse–Kelley set theory"><span class="ilh-page"><a href="/w/index.php?title=Morse%E2%80%93Kelley%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="Morse–Kelley集合论（页面不存在）">Morse–Kelley</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory" class="extiw" title="en:Morse–Kelley set theory"><span lang="en" dir="auto">Morse–Kelley set theory</span></a></span>）</span></span></li></ul></li> <li><span class="ilh-all" data-orig-title="克里普克–普拉特克集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Kripke–Platek set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%8B%E9%87%8C%E6%99%AE%E5%85%8B%E2%80%93%E6%99%AE%E6%8B%89%E7%89%B9%E5%85%8B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="克里普克–普拉特克集合论（页面不存在）">克里普克–普拉特克</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory" class="extiw" title="en:Kripke–Platek set theory"><span lang="en" dir="auto">Kripke–Platek set theory</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="塔斯基–格罗滕迪克集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tarski–Grothendieck set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%A1%94%E6%96%AF%E5%9F%BA%E2%80%93%E6%A0%BC%E7%BD%97%E6%BB%95%E8%BF%AA%E5%85%8B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="塔斯基–格罗滕迪克集合论（页面不存在）">塔斯基–格罗滕迪克</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory" class="extiw" title="en:Tarski–Grothendieck set theory"><span lang="en" dir="auto">Tarski–Grothendieck set theory</span></a></span>）</span></span></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><span class="ilh-all" data-orig-title="集合论悖论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Paradoxes of set theory"><span class="ilh-page"><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA%E6%82%96%E8%AE%BA&action=edit&redlink=1" class="new" title="集合论悖论（页面不存在）">悖论</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Paradoxes_of_set_theory" class="extiw" title="en:Paradoxes of set theory"><span lang="en" dir="auto">Paradoxes of set theory</span></a></span>）</span></span></li><li>问题</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BD%97%E7%B4%A0%E6%82%96%E8%AE%BA" title="罗素悖论">罗素悖论</a></li> <li><a href="/wiki/%E8%98%87%E6%96%AF%E6%9E%97%E5%95%8F%E9%A1%8C" title="蘇斯林問題">蘇斯林問題</a></li> <li><a href="/wiki/ZFC%E7%B3%BB%E7%B5%B1%E7%84%A1%E6%B3%95%E7%A2%BA%E5%AE%9A%E7%9A%84%E5%91%BD%E9%A1%8C%E5%88%97%E8%A1%A8" title="ZFC系統無法確定的命題列表">ZFC系統無法確定的命題列表</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:%E9%9B%86%E5%90%88%E8%AB%96%E8%80%85" title="Category:集合論者">集合論者</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%9A%E4%BC%AF%E6%8B%89%E7%BD%95%C2%B7%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94" title="亚伯拉罕·弗兰克尔">亚伯拉罕·弗兰克尔</a></li> <li><a href="/wiki/%E4%BC%AF%E7%89%B9%E5%85%B0%C2%B7%E7%BD%97%E7%B4%A0" title="伯特兰·罗素">伯特兰·罗素</a></li> <li><a href="/wiki/%E6%81%A9%E6%96%AF%E7%89%B9%C2%B7%E7%AD%96%E6%A2%85%E6%B4%9B" title="恩斯特·策梅洛">恩斯特·策梅洛</a></li> <li><a href="/wiki/%E6%A0%BC%E5%A5%A5%E5%B0%94%E6%A0%BC%C2%B7%E5%BA%B7%E6%89%98%E5%B0%94" title="格奥尔格·康托尔">格奥尔格·康托尔</a></li> <li><a href="/wiki/%E7%BA%A6%E7%BF%B0%C2%B7%E5%86%AF%C2%B7%E8%AF%BA%E4%BC%8A%E6%9B%BC" class="mw-redirect" title="约翰·冯·诺伊曼">约翰·冯·诺伊曼</a></li> <li><a href="/wiki/%E5%BA%93%E5%B0%94%E7%89%B9%C2%B7%E5%93%A5%E5%BE%B7%E5%B0%94" title="库尔特·哥德尔">库尔特·哥德尔</a></li> <li><a href="/wiki/%E7%9B%A7%E8%8F%B2%E7%89%B9%C2%B7%E6%BE%A4%E5%BE%B7" class="mw-redirect" title="盧菲特·澤德">盧菲特·澤德</a></li> <li><span class="ilh-all" data-orig-title="保罗·贝尔奈斯" data-lang-code="en" data-lang-name="英语" data-foreign-title="Paul Bernays"><span class="ilh-page"><a href="/w/index.php?title=%E4%BF%9D%E7%BD%97%C2%B7%E8%B4%9D%E5%B0%94%E5%A5%88%E6%96%AF&action=edit&redlink=1" class="new" title="保罗·贝尔奈斯（页面不存在）">保罗·贝尔奈斯</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Paul_Bernays" class="extiw" title="en:Paul Bernays"><span lang="en" dir="auto">Paul Bernays</span></a></span>）</span></span></li> <li><a href="/wiki/%E4%BF%9D%E7%BD%97%C2%B7%E5%AF%87%E6%81%A9" title="保罗·寇恩">保罗·寇恩</a></li> <li><a href="/wiki/%E7%90%86%E6%9F%A5%E5%BE%B7%C2%B7%E6%88%B4%E5%BE%B7%E9%87%91" title="理查德·戴德金">理查德·戴德金</a></li> <li><a href="/wiki/%E6%89%98%E9%A9%AC%E4%BB%80%C2%B7%E8%80%B6%E8%B5%AB" title="托马什·耶赫">托马什·耶赫</a></li> <li><a href="/wiki/%E5%A8%81%E6%8B%89%E5%BE%B7%C2%B7%E8%8C%83%C2%B7%E5%A5%A5%E6%9B%BC%C2%B7%E8%92%AF%E5%9B%A0" title="威拉德·范·奥曼·蒯因">威拉德·蒯因</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"></div><div role="navigation" class="navbox" aria-labelledby="主要的数学领域" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E6%95%B0%E5%AD%A6%E9%A2%86%E5%9F%9F" title="Template:数学领域"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E6%95%B0%E5%AD%A6%E9%A2%86%E5%9F%9F" title="Template talk:数学领域"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E6%95%B0%E5%AD%A6%E9%A2%86%E5%9F%9F" title="Special:编辑页面/Template:数学领域"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="主要的数学领域" style="font-size:110%;margin:0 5em">主要的<a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">数学</a><a href="/wiki/%E6%95%B0%E5%AD%A6%E9%A2%86%E5%9F%9F" title="数学领域">领域</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%8F%B2" title="数学史">历史</a></li> <li><span class="ilh-all" data-orig-title="数学纲要" data-lang-code="en" data-lang-name="英语" data-foreign-title="Outline of mathematics"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B0%E5%AD%A6%E7%BA%B2%E8%A6%81&action=edit&redlink=1" class="new" title="数学纲要（页面不存在）">纲要</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Outline_of_mathematics" class="extiw" title="en:Outline of mathematics"><span lang="en" dir="auto">Outline of mathematics</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="数学主题列表" data-lang-code="en" data-lang-name="英语" data-foreign-title="Lists of mathematics topics"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B0%E5%AD%A6%E4%B8%BB%E9%A2%98%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="数学主题列表（页面不存在）">列表</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Lists_of_mathematics_topics" class="extiw" title="en:Lists of mathematics topics"><span lang="en" dir="auto">Lists of mathematics topics</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E7%AC%A6%E5%8F%B7%E8%A1%A8" title="数学符号表">符号表</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%9F%BA%E7%A1%80" title="数学基础">数学基础</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%8C%83%E7%95%B4%E8%AE%BA" title="范畴论">范畴论</a></li> <li><a class="mw-selflink selflink">集合论</a></li> <li><a href="/wiki/%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91" title="数理逻辑">数理逻辑</a></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%93%B2%E5%AD%A6" title="数学哲学">数学哲学</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="12" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Math.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Math.svg/80px-Math.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Math.svg/120px-Math.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Math.svg/160px-Math.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%BB%A3%E6%95%B0" title="代数">代数</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0" title="抽象代数">抽象</a></li> <li><a href="/wiki/%E4%BA%A4%E6%8F%9B%E4%BB%A3%E6%95%B8" title="交換代數">交換</a></li> <li><a href="/wiki/%E7%BE%A4%E8%AE%BA" title="群论">群论</a></li> <li><a href="/wiki/%E5%88%9D%E7%AD%89%E4%BB%A3%E6%95%B8" title="初等代數">初等代數</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a></li> <li><a href="/wiki/%E5%A4%9A%E9%87%8D%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="多重线性代数">多重线性代数</a></li> <li><a href="/wiki/%E6%B3%9B%E4%BB%A3%E6%95%B0" title="泛代数">泛代数</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90" title="数学分析">数学分析</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86" class="mw-redirect" title="微积分">微积分</a></li> <li><a href="/wiki/%E5%AE%9E%E5%8F%98%E5%87%BD%E6%95%B0%E8%AE%BA" title="实变函数论">实变函数</a></li> <li><a href="/wiki/%E8%A4%87%E5%88%86%E6%9E%90" title="複分析">复变函数</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a></li> <li><a href="/wiki/%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="泛函分析">泛函分析</a></li> <li><a href="/wiki/%E8%AA%BF%E5%92%8C%E5%88%86%E6%9E%90" title="調和分析">調和分析</a></li> <li><a href="/wiki/%E5%82%85%E7%AB%8B%E5%8F%B6%E5%88%86%E6%9E%90" class="mw-redirect" title="傅立叶分析">傅立葉分析</a></li> <li><a href="/wiki/%E5%87%A0%E4%BD%95%E5%88%86%E6%9E%90" title="几何分析">几何分析</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6" title="离散数学">离散数学</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BB%84%E5%90%88%E6%95%B0%E5%AD%A6" title="组合数学">组合数学</a></li> <li><a href="/wiki/%E5%9B%BE%E8%AE%BA" title="图论">图论</a></li> <li><a href="/wiki/%E5%BA%8F%E7%90%86%E8%AE%BA" title="序理论">序理论</a></li> <li><a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AE%BA" title="博弈论">博弈论</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6" title="几何学">几何学</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BB%A3%E6%95%B0%E5%87%A0%E4%BD%95" title="代数几何">代数几何</a></li> <li><a href="/wiki/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95" title="解析几何">解析几何</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="微分几何">微分几何</a></li> <li><a href="/wiki/%E7%A6%BB%E6%95%A3%E5%87%A0%E4%BD%95%E5%AD%A6" title="离散几何学">离散几何学</a></li> <li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何">欧几里得几何</a></li> <li><a href="/wiki/%E9%9D%9E%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="非欧几里得几何">非欧几里得几何</a></li> <li><a href="/wiki/%E6%9C%89%E9%99%90%E5%B9%BE%E4%BD%95%E5%AD%B8" title="有限幾何學">有限几何学</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B0%E8%AE%BA" title="数论">数论</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%AE%97%E6%9C%AF" title="算术">算术</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B8%E6%95%B8%E8%AB%96" title="代數數論">代數數論</a></li> <li><a href="/wiki/%E8%A7%A3%E6%9E%90%E6%95%B0%E8%AE%BA" title="解析数论">解析数论</a></li> <li><a href="/wiki/%E5%87%A0%E4%BD%95%E6%95%B0%E8%AE%BA" title="几何数论">几何数论</a></li> <li><a href="/wiki/%E7%AE%97%E6%9C%AF%E5%87%A0%E4%BD%95" title="算术几何">算术几何</a></li> <li><a href="/wiki/%E4%B8%A2%E7%95%AA%E5%9B%BE%E5%87%A0%E4%BD%95" class="mw-redirect" title="丢番图几何">丢番图几何</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%8B%93%E6%89%91%E5%AD%A6" title="拓扑学">拓扑学</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%82%B9%E9%9B%86%E6%8B%93%E6%89%91%E5%AD%A6" title="点集拓扑学">点集拓扑</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B0%E6%8B%93%E6%89%91" title="代数拓扑">代数拓扑</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E6%8B%93%E6%89%91" title="微分拓扑">微分拓扑</a></li> <li><a href="/wiki/%E5%87%A0%E4%BD%95%E6%8B%93%E6%89%91%E5%AD%A6" title="几何拓扑学">几何拓扑</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%AD%A6" title="统计学">统计学</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%B5%8B%E5%BA%A6" title="测度">测度与概率</a></li> <li><a href="/wiki/%E6%95%B0%E7%90%86%E7%BB%9F%E8%AE%A1%E5%AD%A6" title="数理统计学">数理统计学</a></li> <li><a href="/wiki/%E6%95%B0%E6%8D%AE%E7%A7%91%E5%AD%A6" title="数据科学">数据科学</a></li> <li><a href="/wiki/%E6%8E%A8%E8%AB%96%E7%B5%B1%E8%A8%88%E5%AD%B8" title="推論統計學">统计推断</a></li> <li><a href="/wiki/%E8%BF%B4%E6%AD%B8%E5%88%86%E6%9E%90" title="迴歸分析">迴歸分析</a></li> <li><a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%AD%A6%E4%B9%A0%E7%90%86%E8%AE%BA" title="统计学习理论">统计学习理论</a></li> <li><a href="/wiki/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0" title="机器学习">机器学习</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a></li> <li><a href="/wiki/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84" title="数据结构">数据结构</a>与<a href="/wiki/%E7%AE%97%E6%B3%95" title="算法">算法</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%BF%90%E7%AE%97%E6%95%B0%E5%AD%A6" title="运算数学">计算数学</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%A7%91%E5%AD%A6" title="计算机科学">计算机科学</a></li> <li><a href="/wiki/%E8%AE%A1%E7%AE%97%E7%90%86%E8%AE%BA" title="计算理论">计算理论</a></li> <li><a href="/wiki/%E6%95%B0%E5%80%BC%E5%88%86%E6%9E%90" title="数值分析">数值分析</a></li> <li><a href="/wiki/%E6%9C%80%E4%BC%98%E5%8C%96" title="最优化">最优化</a></li> <li><a href="/wiki/%E8%A8%88%E7%AE%97%E6%A9%9F%E4%BB%A3%E6%95%B8%E7%B3%BB%E7%B5%B1" title="計算機代數系統">计算机代数</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%BA%94%E7%94%A8%E6%95%B0%E5%AD%A6" title="应用数学">应用数学</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%8E%A7%E5%88%B6%E8%AE%BA" title="控制论">控制论</a></li> <li><a href="/wiki/%E4%BF%A1%E6%81%AF%E8%AE%BA" title="信息论">信息论</a></li> <li><a href="/wiki/%E8%AE%A1%E7%AE%97%E5%8C%96%E5%AD%A6" title="计算化学">计算化学</a></li> <li><a href="/wiki/%E6%95%B8%E7%90%86%E7%94%9F%E7%89%A9%E5%AD%B8" title="數理生物學">数理生物学</a></li> <li><a href="/wiki/%E6%95%B0%E7%90%86%E7%BB%8F%E6%B5%8E%E5%AD%A6" title="数理经济学">数理经济学</a></li> <li><a href="/wiki/%E8%AE%A1%E9%87%8F%E7%BB%8F%E6%B5%8E%E5%AD%A6" title="计量经济学">计量经济学</a></li> <li><a href="/wiki/%E6%95%B8%E7%90%86%E9%87%91%E8%9E%8D%E5%AD%B8" title="數理金融學">數理金融學</a></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%BF%83%E7%90%86%E5%AD%A6" title="数学心理学">数学心理学</a></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E7%89%A9%E7%90%86" title="数学物理">数学物理学</a></li> <li><a href="/wiki/%E7%94%9F%E7%89%A9%E7%B5%B1%E8%A8%88%E5%AD%B8" title="生物統計學">生物統計學</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其它</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%A8%9B%E6%A8%82%E6%95%B8%E5%AD%B8" title="娛樂數學">娱乐数学</a></li> <li><span class="ilh-all" data-orig-title="数学与艺术" data-lang-code="en" data-lang-name="英语" data-foreign-title="Mathematics and art"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B0%E5%AD%A6%E4%B8%8E%E8%89%BA%E6%9C%AF&action=edit&redlink=1" class="new" title="数学与艺术（页面不存在）">数学与艺术</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Mathematics_and_art" class="extiw" title="en:Mathematics and art"><span lang="en" dir="auto">Mathematics and art</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E6%95%99%E8%82%B2" title="数学教育">数学教育</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">注释</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li>数学的领域也可根据“<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AD%A6%E7%A7%91%E5%88%86%E7%B1%BB%E6%A0%87%E5%87%86" title="数学学科分类标准">MSC分类标准</a>”或“<a href="/w/index.php?title=%E4%B8%AD%E5%9B%BD%E5%AD%A6%E7%A7%91%E5%88%86%E7%B1%BB%E5%9B%BD%E5%AE%B6%E6%A0%87%E5%87%86/110&action=edit&redlink=1" class="new" title="中国学科分类国家标准/110（页面不存在）">中国学科分类国家标准</a>”进行分类。</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span typeof="mw:File"><span title="分类"><img alt="分类" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:%E6%95%B0%E5%AD%A6" title="Category:数学">分类</a></b></li> <li><span typeof="mw:File"><span title="主题"><img alt="主题" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Portal:%E6%95%B0%E5%AD%A6" title="Portal:数学">主题</a></b></li> <li><span typeof="mw:File"><span title="共享资源页面"><img alt="共享资源页面" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">共享资源</a></b></li> <li><span typeof="mw:File"><span title="专题"><img alt="专题" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span><b><a href="/wiki/WikiProject:%E6%95%B0%E5%AD%A6" title="WikiProject:数学">专题</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"></div><div role="navigation" class="navbox" aria-labelledby="逻辑" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E9%80%BB%E8%BE%91" title="Template:逻辑"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E9%80%BB%E8%BE%91" title="Template talk:逻辑"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E9%80%BB%E8%BE%91" title="Special:编辑页面/Template:逻辑"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="逻辑" style="font-size:110%;margin:0 5em"><a href="/wiki/%E9%80%BB%E8%BE%91" title="逻辑">逻辑</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";"><div id="概要" style="font-size:110%;margin:0 5em">概要</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">学术领域</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="argumentation scheme" data-lang-code="en" data-lang-name="英语" data-foreign-title="Argumentation scheme"><span class="ilh-page"><a href="/w/index.php?title=Argumentation_scheme&action=edit&redlink=1" class="new" title="Argumentation scheme（页面不存在）">辩论法</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Argumentation_scheme" class="extiw" title="en:Argumentation scheme"><span lang="en" dir="auto">Argumentation scheme</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%83%B9%E5%80%BC%E8%AB%96" title="價值論">价值论</a></li> <li><a href="/wiki/%E6%89%B9%E5%88%A4%E6%80%A7%E6%80%9D%E7%BB%B4" title="批判性思维">审辩式思维</a></li> <li><a href="/wiki/%E5%8F%AF%E8%AE%A1%E7%AE%97%E6%80%A7%E7%90%86%E8%AE%BA" title="可计算性理论">可计算性理论</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E7%9A%84%E8%AF%AD%E4%B9%89" class="mw-redirect" title="逻辑的语义">形式语义学</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E5%8F%B2" title="逻辑史">逻辑史</a></li> <li><a href="/wiki/%E9%9D%9E%E5%BD%A2%E5%BC%8F%E9%80%BB%E8%BE%91" title="非形式逻辑">非形式逻辑</a></li> <li><a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E9%80%BB%E8%BE%91" title="计算机逻辑">计算机逻辑</a></li> <li><a href="/wiki/%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91" title="数理逻辑">数理逻辑</a></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">数学</a></li> <li><span class="ilh-all" data-orig-title="元逻辑" data-lang-code="en" data-lang-name="英语" data-foreign-title="Metalogic"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%83%E9%80%BB%E8%BE%91&action=edit&redlink=1" class="new" title="元逻辑（页面不存在）">元逻辑</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Metalogic" class="extiw" title="en:Metalogic"><span lang="en" dir="auto">Metalogic</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%85%83%E6%95%B0%E5%AD%A6" title="元数学">元数学</a></li> <li><a href="/wiki/%E6%A8%A1%E5%9E%8B%E8%AE%BA" title="模型论">模型论</a></li> <li><a href="/wiki/%E5%93%B2%E5%AD%A6%E9%80%BB%E8%BE%91" title="哲学逻辑">哲学逻辑</a></li> <li><a href="/wiki/%E5%93%B2%E5%AD%A6" title="哲学">哲学</a></li> <li><span class="ilh-all" data-orig-title="逻辑哲学" data-lang-code="en" data-lang-name="英语" data-foreign-title="Philosophy of logic"><span class="ilh-page"><a href="/w/index.php?title=%E9%80%BB%E8%BE%91%E5%93%B2%E5%AD%A6&action=edit&redlink=1" class="new" title="逻辑哲学（页面不存在）">逻辑哲学</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Philosophy_of_logic" class="extiw" title="en:Philosophy of logic"><span lang="en" dir="auto">Philosophy of logic</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%93%B2%E5%AD%A6" title="数学哲学">数学哲学</a></li> <li><a href="/wiki/%E8%AF%81%E6%98%8E%E8%AE%BA" title="证明论">证明论</a></li> <li><a class="mw-selflink selflink">集合论</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">基础概念</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%BA%AF%E5%9B%A0%E6%8E%A8%E7%90%86" title="溯因推理">溯因推理</a></li> <li><a href="/wiki/%E5%88%86%E6%9E%90-%E7%B6%9C%E5%90%88%E5%8D%80%E5%88%A5" title="分析-綜合區別">分析真理</a></li> <li><a href="/wiki/%E4%BA%8C%E5%BE%8B%E8%83%8C%E5%8F%8D" title="二律背反">二律背反</a></li> <li><a href="/wiki/%E5%85%88%E9%AA%8C" class="mw-redirect" title="先验">先验</a></li> <li><a href="/wiki/%E6%BC%94%E7%BB%8E%E6%8E%A8%E7%90%86" title="演绎推理">演绎推理</a></li> <li><a href="/wiki/%E5%AE%9A%E4%B9%89" title="定义">定义</a></li> <li><a href="/wiki/%E6%8F%8F%E8%BF%B0%E9%80%BB%E8%BE%91" title="描述逻辑">描述逻辑</a></li> <li><a href="/wiki/%E5%BD%92%E7%BA%B3%E6%8E%A8%E7%90%86" title="归纳推理">归纳推理</a></li> <li><a href="/wiki/%E6%8E%A8%E8%AE%BA" title="推论">推论</a> <ul><li><span class="ilh-all" data-orig-title="推理方式" data-lang-code="d" data-lang-name="維基數據" data-foreign-title="Q126723190"><span class="ilh-page"><a href="/w/index.php?title=%E6%8E%A8%E7%90%86%E6%96%B9%E5%BC%8F&action=edit&redlink=1" class="new" title="推理方式（页面不存在）">推理方式</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">維基數據所列</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://www.wikidata.org/wiki/Q126723190" class="extiw" title="d:Q126723190"><span lang="d" dir="auto">Q126723190</span></a></span>）</span></span></li></ul></li> <li><a href="/wiki/%E8%95%B4%E6%B6%B5" title="蕴涵">蕴涵</a></li> <li><span class="ilh-all" data-orig-title="模式 (邏輯)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Logical form"><span class="ilh-page"><a href="/w/index.php?title=%E6%A8%A1%E5%BC%8F_(%E9%82%8F%E8%BC%AF)&action=edit&redlink=1" class="new" title="模式 (邏輯)（页面不存在）">逻辑形式</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Logical_form" class="extiw" title="en:Logical form"><span lang="en" dir="auto">Logical form</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%82%8F%E8%BC%AF%E7%9C%9F%E7%90%86" title="邏輯真理">逻辑真理</a></li> <li><a href="/wiki/%E5%90%8D%E7%A7%B0" title="名称">名称</a></li> <li><a href="/wiki/%E5%85%85%E5%88%86%E5%BF%85%E8%A6%81%E6%9D%A1%E4%BB%B6" title="充分必要条件">充分必要条件</a></li> <li><a href="/wiki/%E6%82%96%E8%AE%BA" title="悖论">悖论</a></li> <li><a href="/wiki/%E5%8F%AF%E8%83%BD%E4%B8%96%E7%95%8C" title="可能世界">可能世界</a></li> <li><a href="/wiki/%E5%89%8D%E8%A8%AD" title="前設">前設</a> <ul><li><a href="/wiki/%E5%81%87%E8%AF%B4" title="假说">假说</a></li></ul></li> <li><a href="/wiki/%E6%A6%82%E7%8E%87" title="概率">机率</a></li> <li><a href="/wiki/%E7%90%86%E6%99%BA" title="理智">理智</a></li> <li><a href="/wiki/%E7%90%86%E6%99%BA" title="理智">推理</a></li> <li><a href="/wiki/%E5%8F%83%E8%80%83" title="參考">指涉</a></li> <li><a href="/wiki/%E8%AF%AD%E4%B9%89%E5%AD%A6" title="语义学">语义学</a></li> <li><a href="/wiki/%E5%91%BD%E9%A2%98" title="命题">命题</a></li> <li><a href="/wiki/%E4%B8%A5%E6%A0%BC%E6%9D%A1%E4%BB%B6" title="严格条件">严格条件</a></li> <li><span class="ilh-all" data-orig-title="代入 (邏輯)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Substitution (logic)"><span class="ilh-page"><a href="/w/index.php?title=%E4%BB%A3%E5%85%A5_(%E9%82%8F%E8%BC%AF)&action=edit&redlink=1" class="new" title="代入 (邏輯)（页面不存在）">代换</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Substitution_(logic)" class="extiw" title="en:Substitution (logic)"><span lang="en" dir="auto">Substitution (logic)</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%AF%AD%E6%B3%95" title="语法">语法</a></li> <li><a href="/wiki/%E7%9C%9F%E7%90%86" title="真理">真理</a></li> <li><a href="/wiki/%E7%9C%9F%E5%80%BC" title="真值">真值</a></li> <li><a href="/wiki/%E6%9C%89%E6%95%88%E6%80%A7" title="有效性">有效性</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";"><div id="哲学逻辑" style="font-size:110%;margin:0 5em"><a href="/wiki/%E5%93%B2%E5%AD%A6%E9%80%BB%E8%BE%91" title="哲学逻辑">哲学逻辑</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%89%B9%E5%88%A4%E6%80%A7%E6%80%9D%E7%BB%B4" title="批判性思维">审辩式思维</a><br />和<a href="/wiki/%E9%9D%9E%E5%BD%A2%E5%BC%8F%E9%80%BB%E8%BE%91" title="非形式逻辑">非形式逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%93%B2%E5%AD%B8%E5%88%86%E6%9E%90" title="哲學分析">分析</a></li> <li><a href="/wiki/%E6%AD%A7%E7%BE%A9%E6%80%A7" class="mw-redirect" title="歧義性">歧义性</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E8%AE%BA%E8%AF%81" title="逻辑论证">论证</a></li> <li><a href="/wiki/%E4%BF%A1%E4%BB%B0" title="信仰">信仰</a></li> <li><a href="/wiki/%E5%81%8F%E8%A6%8B" title="偏見">偏见</a></li> <li><a href="/wiki/%E5%85%AC%E4%BF%A1%E5%8A%9B" title="公信力">公信力</a></li> <li><a href="/wiki/%E8%AF%81%E6%8D%AE" class="mw-redirect" title="证据">证据</a></li> <li><a href="/wiki/%E8%A7%A3%E9%87%8A" class="mw-redirect" title="解释">解释</a></li> <li><a href="/wiki/%E8%A7%A3%E9%87%8A%E5%8A%9B" title="解释力">解释力</a></li> <li><a href="/wiki/%E4%BA%8B%E5%AF%A6" title="事實">事实</a></li> <li><a href="/wiki/%E8%B0%AC%E8%AE%BA" class="mw-redirect" title="谬论">谬论</a></li> <li><span class="ilh-all" data-orig-title="探究" data-lang-code="en" data-lang-name="英语" data-foreign-title="Inquiry"><span class="ilh-page"><a href="/w/index.php?title=%E6%8E%A2%E7%A9%B6&action=edit&redlink=1" class="new" title="探究（页面不存在）">探究</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Inquiry" class="extiw" title="en:Inquiry"><span lang="en" dir="auto">Inquiry</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%84%8F%E8%A7%81" title="意见">意见</a></li> <li><a href="/wiki/%E5%A5%A5%E5%8D%A1%E5%A7%86%E5%89%83%E5%88%80" title="奥卡姆剃刀">奥卡姆剃刀</a></li> <li><a href="/wiki/%E5%89%8D%E6%8F%90" title="前提">前提</a></li> <li><a href="/wiki/%E6%94%BF%E6%B2%BB%E5%AE%A3%E4%BC%A0" title="政治宣传">政治宣传</a></li> <li><a href="/wiki/%E5%AF%A9%E6%85%8E" title="審慎">审慎</a></li> <li><a href="/wiki/%E6%8E%A8%E7%90%86" title="推理">推理</a></li> <li><a href="/wiki/%E5%85%B3%E8%81%94" class="mw-redirect" title="关联">关联</a></li> <li><a href="/wiki/%E4%BF%AE%E8%BE%9E%E5%AD%A6" title="修辞学">修辞学</a></li> <li><a href="/wiki/%E4%B8%A5%E8%B0%A8_(%E6%95%B0%E5%AD%A6)" title="严谨 (数学)">严谨</a></li> <li><a href="/wiki/%E5%90%AB%E7%B3%8A" title="含糊">含糊</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%BC%94%E7%BB%8E%E6%8E%A8%E7%90%86" title="演绎推理">演绎推理</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%95%B0%E5%AD%A6%E7%BB%93%E6%9E%84%E4%B8%BB%E4%B9%89" class="mw-redirect" title="数学结构主义">结构主义</a></li> <li><a href="/wiki/%E5%8F%8C%E9%9D%A2%E7%9C%9F%E7%90%86%E8%AF%B4" title="双面真理说">双面真理说</a></li> <li><span class="ilh-all" data-orig-title="fictionalism" data-lang-code="en" data-lang-name="英语" data-foreign-title="Fictionalism"><span class="ilh-page"><a href="/w/index.php?title=Fictionalism&action=edit&redlink=1" class="new" title="Fictionalism（页面不存在）">虚构主义</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Fictionalism" class="extiw" title="en:Fictionalism"><span lang="en" dir="auto">Fictionalism</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%9C%89%E9%99%90%E4%B8%BB%E7%BE%A9" title="有限主義">有限主义</a></li> <li><a href="/wiki/%E5%BD%A2%E5%BC%8F%E4%B8%BB%E4%B9%89" title="形式主义">形式主义</a></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E7%9B%B4%E8%A7%89%E4%B8%BB%E4%B9%89" class="mw-redirect" title="数学直觉主义">直觉主义</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E5%8E%9F%E5%AD%90%E8%AE%BA" title="逻辑原子论">逻辑原子论</a></li> <li><a href="/wiki/%E9%82%8F%E8%BC%AF%E4%B8%BB%E7%BE%A9" title="邏輯主義">逻辑主义</a></li> <li><a href="/wiki/%E5%94%AF%E5%90%8D%E8%AB%96" title="唯名論">唯名论</a></li> <li><a href="/wiki/%E5%AE%9E%E7%94%A8%E4%B8%BB%E4%B9%89" title="实用主义">实用主义</a></li> <li><a href="/wiki/%E5%94%AF%E5%AF%A6%E8%AB%96" class="mw-redirect" title="唯實論">唯实论</a> <ul><li><a href="/wiki/%E6%9F%8F%E6%8B%89%E5%9C%96%E5%AF%A6%E5%9C%A8%E8%AB%96" title="柏拉圖實在論">柏拉圖實在論</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";"><div id="元逻辑（英语：-&#123;metalogic&#125;-）和元数学" style="font-size:110%;margin:0 5em"><span class="ilh-all" data-orig-title="元逻辑" data-lang-code="en" data-lang-name="英语" data-foreign-title="metalogic"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%83%E9%80%BB%E8%BE%91&action=edit&redlink=1" class="new" title="元逻辑（页面不存在）">元逻辑</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/metalogic" class="extiw" title="en:metalogic"><span lang="en" dir="auto">metalogic</span></a></span>）</span></span>和<a href="/wiki/%E5%85%83%E6%95%B0%E5%AD%A6" title="元数学">元数学</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94%E5%AE%9A%E7%90%86" title="康托尔定理">康托尔定理</a></li> <li><a href="/wiki/%E5%8F%AF%E5%88%A4%E5%AE%9A%E6%80%A7" title="可判定性">可判定性</a></li> <li><a href="/wiki/%E9%82%B1%E5%A5%87-%E5%9B%BE%E7%81%B5%E8%AE%BA%E9%A2%98" title="邱奇-图灵论题">邱奇-图灵论题</a></li> <li><a href="/wiki/%E4%B8%80%E8%87%B4%E6%80%A7_(%E9%82%8F%E8%BC%AF)" title="一致性 (邏輯)">一致性</a></li> <li><span class="ilh-all" data-orig-title="有效方法" data-lang-code="en" data-lang-name="英语" data-foreign-title="Effective method"><span class="ilh-page"><a href="/w/index.php?title=%E6%9C%89%E6%95%88%E6%96%B9%E6%B3%95&action=edit&redlink=1" class="new" title="有效方法（页面不存在）">有效方法</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Effective_method" class="extiw" title="en:Effective method"><span lang="en" dir="auto">Effective method</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%9F%BA%E7%A1%80" title="数学基础">数学基础</a></li> <li><a href="/wiki/%E5%93%A5%E5%BE%B7%E5%B0%94%E5%AE%8C%E5%A4%87%E6%80%A7%E5%AE%9A%E7%90%86" title="哥德尔完备性定理">哥德尔完备性定理</a></li> <li><a href="/wiki/%E5%93%A5%E5%BE%B7%E5%B0%94%E4%B8%8D%E5%AE%8C%E5%A4%87%E5%AE%9A%E7%90%86" title="哥德尔不完备定理">哥德尔不完备定理</a></li> <li><a href="/wiki/%E5%8F%AF%E9%9D%A0%E6%80%A7%E5%AE%9A%E7%90%86" title="可靠性定理">可靠性</a></li> <li><a href="/wiki/%E5%AE%8C%E5%A4%87%E6%80%A7" title="完备性">完备性</a></li> <li><a href="/wiki/%E5%8F%AF%E5%88%A4%E5%AE%9A%E6%80%A7" title="可判定性">可判定性</a></li> <li><a href="/wiki/%E8%A7%A3%E9%87%8B_(%E9%82%8F%E8%BC%AF)" title="解釋 (邏輯)">解释</a></li> <li><a href="/wiki/%E5%8B%92%E6%96%87%E6%B5%B7%E5%A7%86%E2%80%93%E6%96%AF%E7%A7%91%E4%BC%A6%E5%AE%9A%E7%90%86" title="勒文海姆–斯科伦定理">勒文海姆–斯科伦定理</a></li> <li><a href="/wiki/%E5%85%83%E5%AE%9A%E7%90%86" title="元定理">元定理</a></li> <li><a href="/wiki/%E5%B8%83%E5%B0%94%E5%8F%AF%E6%BB%A1%E8%B6%B3%E6%80%A7%E9%97%AE%E9%A2%98" title="布尔可满足性问题">可满足性</a></li> <li><a href="/wiki/%E7%8D%A8%E7%AB%8B%E6%80%A7_(%E6%95%B8%E7%90%86%E9%82%8F%E8%BC%AF)" title="獨立性 (數理邏輯)">独立性</a></li> <li><a href="/wiki/%E7%B1%BB%E5%9E%8B-%E8%AE%B0%E5%8F%B7%E5%8C%BA%E5%88%AB" class="mw-redirect" title="类型-记号区别">类型-记号区别</a></li> <li><a href="/wiki/%E4%BD%BF%E7%94%A8-%E6%8F%90%E5%8F%8A%E5%8C%BA%E5%88%AB" class="mw-redirect" title="使用-提及区别">使用-提及区别</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";"><div id="非传统逻辑" style="font-size:110%;margin:0 5em">非传统逻辑</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%A8%A1%E6%80%81%E9%80%BB%E8%BE%91" title="模态逻辑">模态逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/w/index.php?title=%E7%9C%9F%E6%80%A7%E6%A8%A1%E6%80%81&action=edit&redlink=1" class="new" title="真性模态（页面不存在）">真性逻辑</a></li> <li><a href="/w/index.php?title=%E6%A8%A1%E6%80%81%E8%BF%90%E7%AE%97%E7%AC%A6&action=edit&redlink=1" class="new" title="模态运算符（页面不存在）">价值逻辑</a></li> <li><a href="/wiki/%E9%81%93%E4%B9%89%E9%80%BB%E8%BE%91" title="道义逻辑">道义逻辑</a></li> <li><span class="ilh-all" data-orig-title="信念逻辑" data-lang-code="en" data-lang-name="英语" data-foreign-title="Doxastic logic"><span class="ilh-page"><a href="/w/index.php?title=%E4%BF%A1%E5%BF%B5%E9%80%BB%E8%BE%91&action=edit&redlink=1" class="new" title="信念逻辑（页面不存在）">信念逻辑</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Doxastic_logic" class="extiw" title="en:Doxastic logic"><span lang="en" dir="auto">Doxastic logic</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%AE%A4%E8%AF%86%E9%80%BB%E8%BE%91" title="认识逻辑">认识逻辑</a></li> <li><a href="/wiki/%E6%97%B6%E9%97%B4%E9%80%BB%E8%BE%91" title="时间逻辑">时间逻辑</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B0%E5%AD%A6%E7%9B%B4%E8%A7%89%E4%B8%BB%E4%B9%89" class="mw-redirect" title="数学直觉主义">直觉主义</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9B%B4%E8%A7%89%E4%B8%BB%E4%B9%89%E9%80%BB%E8%BE%91" title="直觉主义逻辑">直觉主义逻辑</a></li> <li><a href="/wiki/%E7%BB%93%E6%9E%84%E5%88%86%E6%9E%90" title="结构分析">结构分析</a></li> <li><span class="ilh-all" data-orig-title="Heyting算术" data-lang-code="en" data-lang-name="英语" data-foreign-title="Heyting arithmetic"><span class="ilh-page"><a href="/w/index.php?title=Heyting%E7%AE%97%E6%9C%AF&action=edit&redlink=1" class="new" title="Heyting算术（页面不存在）">Heyting算术</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Heyting_arithmetic" class="extiw" title="en:Heyting arithmetic"><span lang="en" dir="auto">Heyting arithmetic</span></a></span>）</span></span></li> <li><a href="/wiki/%E7%9B%B4%E8%A7%89%E7%B1%BB%E5%9E%8B%E8%AE%BA" title="直觉类型论">直觉类型论</a></li> <li><span class="ilh-all" data-orig-title="建構式集合論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Constructive set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%BB%BA%E6%A7%8B%E5%BC%8F%E9%9B%86%E5%90%88%E8%AB%96&action=edit&redlink=1" class="new" title="建構式集合論（页面不存在）">建構式集合論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Constructive_set_theory" class="extiw" title="en:Constructive set theory"><span lang="en" dir="auto">Constructive set theory</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%80%BB%E8%BE%91" title="模糊逻辑">模糊逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="真实性程度" data-lang-code="en" data-lang-name="英语" data-foreign-title="Degree of truth"><span class="ilh-page"><a href="/w/index.php?title=%E7%9C%9F%E5%AE%9E%E6%80%A7%E7%A8%8B%E5%BA%A6&action=edit&redlink=1" class="new" title="真实性程度（页面不存在）">真实性程度</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Degree_of_truth" class="extiw" title="en:Degree of truth"><span lang="en" dir="auto">Degree of truth</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%A8%A1%E7%B3%8A%E8%A7%84%E5%88%99" title="模糊规则">模糊规则</a></li> <li><a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集</a></li> <li><a href="/w/index.php?title=%E6%A8%A1%E7%B3%8A%E6%9C%89%E9%99%90%E5%85%83%E7%B4%A0&action=edit&redlink=1" class="new" title="模糊有限元素（页面不存在）">模糊有限元素</a></li> <li><span class="ilh-all" data-orig-title="模糊集合運算" data-lang-code="en" data-lang-name="英语" data-foreign-title="Fuzzy set operations"><span class="ilh-page"><a href="/w/index.php?title=%E6%A8%A1%E7%B3%8A%E9%9B%86%E5%90%88%E9%81%8B%E7%AE%97&action=edit&redlink=1" class="new" title="模糊集合運算（页面不存在）">模糊集合運算</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Fuzzy_set_operations" class="extiw" title="en:Fuzzy set operations"><span lang="en" dir="auto">Fuzzy set operations</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%BA%9A%E7%BB%93%E6%9E%84%E9%80%BB%E8%BE%91" title="亚结构逻辑">亚结构逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BB%93%E6%9E%84%E8%A7%84%E5%88%99" title="结构规则">结构规则</a></li> <li><a href="/wiki/%E7%9B%B8%E5%B9%B2%E9%80%BB%E8%BE%91" title="相干逻辑">相干逻辑</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E9%80%BB%E8%BE%91" title="线性逻辑">线性逻辑</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%AC%A1%E5%8D%8F%E8%B0%83%E9%80%BB%E8%BE%91" title="次协调逻辑">次协调逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8F%8C%E9%9D%A2%E7%9C%9F%E7%90%86%E8%AF%B4" title="双面真理说">双面真理说</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%8F%8F%E8%BF%B0%E9%80%BB%E8%BE%91" title="描述逻辑">描述逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%9C%AC%E4%BD%93%E8%AE%BA_(%E5%93%B2%E5%AD%A6)" title="本体论 (哲学)">本体论</a></li> <li><a href="/wiki/%E6%9C%AC%E4%BD%93%E8%AF%AD%E8%A8%80" title="本体语言">本体语言</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";"><div id="逻辑学家" style="font-size:110%;margin:0 5em"><a href="/wiki/%E9%80%BB%E8%BE%91%E5%AD%A6%E5%AE%B6" class="mw-redirect" title="逻辑学家">逻辑学家</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/w/index.php?title=%E9%9B%85%E4%BC%A6%C2%B7%E7%BD%97%E6%96%AF%C2%B7%E5%AE%89%E5%BE%B7%E9%80%8A&action=edit&redlink=1" class="new" title="雅伦·罗斯·安德逊（页面不存在）">安德逊</a></li> <li><a href="/wiki/%E4%BA%9A%E9%87%8C%E5%A3%AB%E5%A4%9A%E5%BE%B7" title="亚里士多德">亚里斯多德</a></li> <li><a href="/wiki/%E4%BC%8A%E6%9C%AC%C2%B7%E9%AD%AF%E4%B8%96%E5%BE%B7" title="伊本·魯世德">鲁世德</a></li> <li><a href="/wiki/%E4%BC%8A%E6%9C%AC%C2%B7%E8%A5%BF%E9%82%A3" title="伊本·西那">西那</a></li> <li><a href="/w/index.php?title=%E4%BA%9A%E5%8E%86%E5%B1%B1%E5%A4%A7%C2%B7%E8%B4%9D%E6%81%A9&action=edit&redlink=1" class="new" title="亚历山大·贝恩（页面不存在）">贝恩</a></li> <li><a href="/w/index.php?title=%E4%B9%94%E6%81%A9%C2%B7%E5%B7%B4%E5%A8%81%E6%96%AF&action=edit&redlink=1" class="new" title="乔恩·巴威斯（页面不存在）">巴威斯</a></li> <li><a href="/w/index.php?title=%E4%BF%9D%E7%BD%97%C2%B7%E5%8D%9A%E5%86%85%E6%96%AF&action=edit&redlink=1" class="new" title="保罗·博内斯（页面不存在）">博内斯</a></li> <li><a href="/wiki/%E4%B9%94%E6%B2%BB%C2%B7%E5%B8%83%E5%B0%94" title="乔治·布尔">布尔</a></li> <li><a href="/w/index.php?title=%E4%B9%94%E6%B2%BB%C2%B7%E5%B8%83%E5%8B%92%E6%96%AF&action=edit&redlink=1" class="new" title="乔治·布勒斯（页面不存在）">布勒斯</a></li> <li><a href="/wiki/%E6%A0%BC%E5%A5%A5%E5%B0%94%E6%A0%BC%C2%B7%E5%BA%B7%E6%89%98%E5%B0%94" title="格奥尔格·康托尔">康托尔</a></li> <li><a href="/wiki/%E9%B2%81%E9%81%93%E5%A4%AB%C2%B7%E5%8D%A1%E5%B0%94%E7%BA%B3%E6%99%AE" title="鲁道夫·卡尔纳普">卡尔纳普</a></li> <li><a href="/wiki/%E9%98%BF%E9%9A%86%E4%BD%90%C2%B7%E9%82%B1%E5%A5%87" title="阿隆佐·邱奇">邱奇</a></li> <li><a href="/w/index.php?title=%E5%85%8B%E5%90%95%E8%A5%BF%E6%B3%A2&action=edit&redlink=1" class="new" title="克吕西波（页面不存在）">克吕西波</a></li> <li><a href="/wiki/%E5%93%88%E6%96%AF%E5%87%B1%E7%88%BE%C2%B7%E5%8A%A0%E9%87%8C" 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href="/wiki/%E7%B4%A2%E5%B0%94%C2%B7%E9%98%BF%E4%BC%A6%C2%B7%E5%85%8B%E9%87%8C%E6%99%AE%E5%85%8B" title="索尔·阿伦·克里普克">克里普克</a></li> <li><a href="/wiki/%E6%88%88%E7%89%B9%E5%BC%97%E9%87%8C%E5%BE%B7%C2%B7%E8%8E%B1%E5%B8%83%E5%B0%BC%E8%8C%A8" title="戈特弗里德·莱布尼茨">莱布尼兹</a></li> <li><a href="/wiki/%E5%88%A9%E5%A5%A5%E6%B3%A2%E5%BE%B7%C2%B7%E5%8B%92%E6%96%87%E6%B5%B7%E5%A7%86" title="利奥波德·勒文海姆">勒文海姆</a></li> <li><a href="/wiki/%E6%9C%B1%E5%A1%9E%E4%BD%A9%C2%B7%E7%9A%AE%E4%BA%9E%E8%AB%BE" title="朱塞佩·皮亞諾">皮亚诺</a></li> <li><a href="/wiki/%E6%9F%A5%E5%B0%94%E6%96%AF%C2%B7%E6%A1%91%E5%BE%B7%E6%96%AF%C2%B7%E7%9A%AE%E5%B0%94%E5%A3%AB" title="查尔斯·桑德斯·皮尔士">皮尔士</a></li> <li><a href="/wiki/%E5%B8%8C%E6%8B%89%E9%87%8C%C2%B7%E6%80%80%E7%89%B9%E5%93%88%E5%B0%94%C2%B7%E6%99%AE%E7%89%B9%E5%8D%97" title="希拉里·怀特哈尔·普特南">普特南</a></li> <li><a href="/wiki/%E5%A8%81%E6%8B%89%E5%BE%B7%C2%B7%E5%86%AF%C2%B7%E5%A5%A5%E6%9B%BC%C2%B7%E8%92%AF%E5%9B%A0" class="mw-redirect" title="威拉德·冯·奥曼·蒯因">奎因</a></li> <li><a href="/wiki/%E4%BC%AF%E7%89%B9%E5%85%B0%C2%B7%E7%BD%97%E7%B4%A0" title="伯特兰·罗素">罗素</a></li> <li><a href="/w/index.php?title=%E6%81%A9%E6%96%AF%E7%89%B9%C2%B7%E6%96%BD%E7%BD%97%E5%BE%B7&action=edit&redlink=1" class="new" title="恩斯特·施罗德（页面不存在）">施罗德</a></li> <li><a href="/wiki/%E9%82%93%E6%96%AF%C2%B7%E5%8F%B8%E5%90%84%E8%84%B1" title="邓斯·司各脱">司各脱</a></li> <li><a href="/wiki/%E6%89%98%E6%8B%89%E5%B0%94%E5%A4%AB%C2%B7%E6%96%AF%E7%A7%91%E4%BC%A6" title="托拉尔夫·斯科伦">斯科伦</a></li> <li><a href="/wiki/%E9%9B%B7%E8%92%99%C2%B7%E5%8F%B2%E6%85%95%E6%8F%9A" class="mw-redirect" title="雷蒙·史慕揚">史慕扬</a></li> <li><a href="/wiki/%E9%98%BF%E5%B0%94%E5%BC%97%E9%9B%B7%E5%BE%B7%C2%B7%E5%A1%94%E6%96%AF%E5%9F%BA" title="阿尔弗雷德·塔斯基">塔斯基</a></li> <li><a href="/wiki/%E8%89%BE%E4%BC%A6%C2%B7%E5%9B%BE%E7%81%B5" title="艾伦·图灵">图灵</a></li> <li><a href="/wiki/%E9%98%BF%E7%88%BE%E5%BC%97%E9%9B%B7%E5%BE%B7%C2%B7%E8%AB%BE%E6%80%9D%C2%B7%E6%87%B7%E7%89%B9%E9%BB%91%E5%BE%B7" title="阿爾弗雷德·諾思·懷特黑德">怀特黑德</a></li> <li><a href="/wiki/%E5%A5%A5%E5%8D%A1%E5%A7%86%E7%9A%84%E5%A8%81%E5%BB%89" title="奥卡姆的威廉">奥卡姆的威廉</a></li> <li><a href="/wiki/%E8%B7%AF%E5%BE%B7%E7%BB%B4%E5%B8%8C%C2%B7%E7%BB%B4%E7%89%B9%E6%A0%B9%E6%96%AF%E5%9D%A6" title="路德维希·维特根斯坦">维根斯坦</a></li> <li><a href="/wiki/%E6%81%A9%E6%96%AF%E7%89%B9%C2%B7%E7%AD%96%E6%A2%85%E6%B4%9B" title="恩斯特·策梅洛">策梅洛</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";"><div id="列表" style="font-size:110%;margin:0 5em">列表</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">主题</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="逻辑学大纲" data-lang-code="en" data-lang-name="英语" data-foreign-title="Outline of logic"><span class="ilh-page"><a href="/w/index.php?title=%E9%80%BB%E8%BE%91%E5%AD%A6%E5%A4%A7%E7%BA%B2&action=edit&redlink=1" class="new" title="逻辑学大纲（页面不存在）">逻辑概要</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Outline_of_logic" class="extiw" title="en:Outline of logic"><span lang="en" dir="auto">Outline of logic</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="数理逻辑主题列表" data-lang-code="en" data-lang-name="英语" data-foreign-title="List of mathematical logic topics"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91%E4%B8%BB%E9%A2%98%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="数理逻辑主题列表（页面不存在）">数理逻辑</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/List_of_mathematical_logic_topics" class="extiw" title="en:List of mathematical logic topics"><span lang="en" dir="auto">List of mathematical logic topics</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%B8%83%E5%B0%94%E4%BB%A3%E6%95%B0%E4%B8%BB%E9%A2%98%E5%88%97%E8%A1%A8" title="布尔代数主题列表">布尔代数</a></li> <li><span class="ilh-all" data-orig-title="集合論主題列表" data-lang-code="en" data-lang-name="英语" data-foreign-title="List of set theory topics"><span class="ilh-page"><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AB%96%E4%B8%BB%E9%A1%8C%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="集合論主題列表（页面不存在）">集合论</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/List_of_set_theory_topics" class="extiw" title="en:List of set theory topics"><span lang="en" dir="auto">List of set theory topics</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其它</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%80%BB%E8%BE%91%E5%AD%A6%E5%AE%B6" class="mw-redirect" title="逻辑学家">逻辑学家</a></li> <li><span class="ilh-all" data-orig-title="推理规则列表" data-lang-code="en" data-lang-name="英语" data-foreign-title="List of rules of inference"><span class="ilh-page"><a href="/w/index.php?title=%E6%8E%A8%E7%90%86%E8%A7%84%E5%88%99%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="推理规则列表（页面不存在）">推理规则</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/List_of_rules_of_inference" class="extiw" title="en:List of rules of inference"><span lang="en" dir="auto">List of rules of inference</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%82%96%E8%AE%BA%E5%88%97%E8%A1%A8" title="悖论列表">悖论</a></li> <li><a href="/wiki/%E8%AC%AC%E8%AA%A4%E5%88%97%E8%A1%A8" title="謬誤列表">谬论</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E7%AC%A6%E5%8F%B7%E8%A1%A8" title="逻辑符号表">逻辑符号</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";"><div id="常见逻辑符号" style="font-size:110%;margin:0 5em"><a href="/wiki/%E9%80%BB%E8%BE%91%E7%AC%A6%E5%8F%B7%E8%A1%A8" title="逻辑符号表">常见逻辑符号</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><div style="font-size:130%;"> <p><a href="/wiki/%26" title="&">&</a>    <a href="/wiki/%E2%88%A8" class="mw-redirect" title="∨">∨</a>    <a href="/wiki/%C2%AC" class="mw-redirect" title="¬">¬</a>    <a href="/wiki/%E6%B3%A2%E6%B5%AA%E5%8F%B7" class="mw-redirect" title="波浪号">~</a>    <a href="/wiki/%E7%AE%AD%E5%A4%B4" title="箭头">→</a>    <a href="/wiki/%E2%8A%83" class="mw-redirect" title="⊃">⊃</a>    <a href="/wiki/%E8%8B%A5%E4%B8%94%E5%94%AF%E8%8B%A5" class="mw-redirect" title="若且唯若">≡</a>    <a href="/wiki/%E8%B0%A2%E8%B4%B9%E5%B0%94%E7%AB%96%E7%BA%BF" title="谢费尔竖线"><span class="Unicode">|</span></a>    <a href="/wiki/%E5%85%A8%E7%A7%B0%E9%87%8F%E5%8C%96" title="全称量化">∀</a>    <a href="/wiki/%E5%AD%98%E5%9C%A8%E9%87%8F%E5%8C%96" title="存在量化">∃</a>    <a href="/w/index.php?title=%E2%8A%A4&action=edit&redlink=1" class="new" title="⊤（页面不存在）">⊤</a>    <a href="/wiki/%E2%8A%A5" class="mw-redirect" title="⊥">⊥</a>    <a href="/wiki/%E2%8A%A2" class="mw-redirect" title="⊢">⊢</a>    <a href="/wiki/%E2%8A%A8" class="mw-redirect" title="⊨">⊨</a>    <a href="/wiki/%E6%89%80%E4%BB%A5%E7%AC%A6%E8%99%9F" title="所以符號">∴</a>    <a href="/w/index.php?title=%E5%9B%A0%E4%B8%BA%E7%AC%A6%E5%8F%B7&action=edit&redlink=1" class="new" title="因为符号（页面不存在）">∵</a> </p> </div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><div class="hlist"><ul><li>参见：<span class="noviewer" typeof="mw:File"><a href="/wiki/File:Socrates.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" /></a></span> <a href="/wiki/Portal:%E5%93%B2%E5%AD%A6" title="Portal:哲学">哲学主题</a></li><li><span typeof="mw:File"><span title="分类"><img alt="分类" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><a href="/wiki/Category:%E9%82%8F%E8%BC%AF" title="Category:邏輯">分类</a></li></ul></div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"></div><div role="navigation" class="navbox" aria-labelledby="计算机科学的主要领域" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="Template:電腦科學"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="Template talk:電腦科學"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="Special:编辑页面/Template:電腦科學"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="计算机科学的主要领域" style="font-size:110%;margin:0 5em"><a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%A7%91%E5%AD%A6" title="计算机科学">计算机科学</a>的主要领域</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>注：该模板大致遵循<a href="/wiki/ACM_%E7%94%B5%E8%84%91%E5%88%86%E7%B1%BB%E7%B3%BB%E7%BB%9F" title="ACM 电脑分类系统">ACM 电脑分类系统</a>。</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%A1%AC%E4%BB%B6" title="计算机硬件">计算机硬件</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8D%B0%E5%88%B7%E7%94%B5%E8%B7%AF%E6%9D%BF" class="mw-redirect" title="印刷电路板">印刷电路板</a></li> <li><a href="/wiki/%E5%A4%96%E9%83%A8%E8%AE%BE%E5%A4%87" title="外部设备">外部设备</a></li> <li><a href="/wiki/%E9%9B%86%E6%88%90%E7%94%B5%E8%B7%AF" title="集成电路">集成电路</a></li> <li><a href="/wiki/%E8%B6%85%E5%A4%A7%E8%A7%84%E6%A8%A1%E9%9B%86%E6%88%90%E7%94%B5%E8%B7%AF" title="超大规模集成电路">超大规模集成电路</a></li> <li><a href="/wiki/%E7%BB%BF%E8%89%B2%E8%AE%A1%E7%AE%97" title="绿色计算">绿色计算</a></li> <li><a href="/wiki/%E9%9B%BB%E5%AD%90%E8%A8%AD%E8%A8%88%E8%87%AA%E5%8B%95%E5%8C%96" title="電子設計自動化">電子設計自動化</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%B3%BB%E7%BB%9F%E6%9E%B6%E6%9E%84" title="系统架构">系统架构</a>组织</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E7%BB%93%E6%9E%84" title="计算机系统结构">電腦系統架構</a></li> <li><a href="/wiki/%E5%B5%8C%E5%85%A5%E5%BC%8F%E7%B3%BB%E7%BB%9F" title="嵌入式系统">嵌入式系统</a></li> <li><a href="/wiki/%E5%AE%9E%E6%97%B6%E8%AE%A1%E7%AE%97" title="实时计算">实时计算</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C" title="计算机网络">网络</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BD%91%E7%BB%9C%E4%BC%A0%E8%BE%93%E5%8D%8F%E8%AE%AE" class="mw-redirect" title="网络传输协议">网络传输协议</a></li> <li><a href="/wiki/%E8%B7%AF%E7%94%B1" title="路由">路由</a></li> <li><a href="/wiki/%E7%BD%91%E7%BB%9C%E6%8B%93%E6%89%91" title="网络拓扑">网络拓扑</a></li> <li><a href="/wiki/%E7%BD%91%E7%BB%9C%E6%9C%8D%E5%8A%A1" class="mw-redirect" title="网络服务">网络服务</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">软件组织</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9B%B4%E8%AD%AF%E5%99%A8" title="直譯器">直譯器</a></li> <li><a href="/wiki/%E4%B8%AD%E9%97%B4%E4%BB%B6" title="中间件">中间件</a></li> <li><a href="/wiki/%E8%99%9B%E6%93%AC%E6%A9%9F%E5%99%A8" title="虛擬機器">虛擬機器</a></li> <li><a href="/wiki/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F" title="操作系统">操作系统</a></li> <li><a href="/wiki/%E8%BD%AF%E4%BB%B6%E8%B4%A8%E9%87%8F" title="软件质量">软件质量</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80%E7%90%86%E8%AB%96" title="程式語言理論">软件符号</a>和<a href="/wiki/%E8%BD%AF%E4%BB%B6%E5%BC%80%E5%8F%91%E5%B7%A5%E5%85%B7" title="软件开发工具">工具</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BC%96%E7%A8%8B%E8%8C%83%E5%9E%8B" title="编程范型">编程范型</a></li> <li><a href="/wiki/%E7%BC%96%E7%A8%8B%E8%AF%AD%E8%A8%80" title="编程语言">编程语言</a></li> <li><a href="/wiki/%E7%B7%A8%E8%AD%AF%E5%99%A8" title="編譯器">編譯器</a></li> <li><a href="/wiki/%E9%A2%86%E5%9F%9F%E7%89%B9%E5%AE%9A%E8%AF%AD%E8%A8%80" title="领域特定语言">领域特定语言</a></li> <li><a href="/wiki/%E8%BB%9F%E9%AB%94%E6%A1%86%E6%9E%B6" title="軟體框架">軟體框架</a></li> <li><a href="/wiki/%E9%9B%86%E6%88%90%E5%BC%80%E5%8F%91%E7%8E%AF%E5%A2%83" title="集成开发环境">集成开发环境</a></li> <li><a href="/wiki/%E8%BD%AF%E4%BB%B6%E9%85%8D%E7%BD%AE%E7%AE%A1%E7%90%86" title="软件配置管理">软件配置管理</a></li> <li><a href="/wiki/%E5%87%BD%E5%BC%8F%E5%BA%AB" title="函式庫">函式庫</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%BD%AF%E4%BB%B6%E5%BC%80%E5%8F%91" title="软件开发">软件开发</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%BD%AF%E4%BB%B6%E5%BC%80%E5%8F%91%E8%BF%87%E7%A8%8B" title="软件开发过程">软件开发过程</a></li> <li><a href="/wiki/%E9%9C%80%E6%B1%82%E5%88%86%E6%9E%90" title="需求分析">需求分析</a></li> <li><a href="/wiki/%E8%BD%AF%E4%BB%B6%E8%AE%BE%E8%AE%A1" 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