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class="vector-toc-list"> </ul> </li> <li id="toc-引用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#引用"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>引用</span> </div> </a> <ul id="toc-引用-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 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class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Zermelo_aksiomaatika" title="Zermelo aksiomaatika – 爱沙尼亚语" lang="et" hreflang="et" data-title="Zermelo aksiomaatika" 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-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_ensembles_de_Zermelo" title="Théorie des ensembles de Zermelo – 法语" lang="fr" hreflang="fr" data-title="Théorie des ensembles de Zermelo" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_degli_insiemi_di_Zermelo" title="Teoria degli insiemi di Zermelo – 意大利语" lang="it" hreflang="it" data-title="Teoria 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class="vector-body-before-content"> <div class="mw-indicators"> </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"><p><b>策梅洛集合论</b>（德語：<span lang="de">Zermelo-Mengenlehre</span>），设立自<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>在1908年的重要论文，它是现代<a href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">集合论</a>的祖先。它与它的后代有特定的差别，经常被误解并经常被误引用。本文架设最初的公理，带有最初的文本（从德文译成了英文）和编号。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="策梅洛集合论的公理"><span id=".E7.AD.96.E6.A2.85.E6.B4.9B.E9.9B.86.E5.90.88.E8.AE.BA.E7.9A.84.E5.85.AC.E7.90.86"></span>策梅洛集合论的公理</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=1" title="编辑章节：策梅洛集合论的公理"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>公理I</b>。<a href="/wiki/%E5%A4%96%E5%BB%B6%E5%85%AC%E7%90%86" title="外延公理">外延性公理</a>（Axiom der Bestimmtheit）：“如果一个集合<i>M</i>的所有元素也是<i>N</i>的元素，且反之亦然...则<i>M</i> = <i>N</i>。简要的说，所有集合由它所包含的元素确定”。</li> <li><b>公理II</b>。基本集合公理（Axiom der Elementarmengen）：“存在这样的一个集合，即空集<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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span>，它根本不包含元素。如果<i>a</i>是域的任何元素，存在一个集合{<i>a</i>}包含<i>a</i>并只包含<i>a</i>作为元素。如果<i>a</i>和<i>b</i>是域的任何两个元素，总是存在一个集合{<i>a</i>, <i>b</i>}包含<i>a</i>和<i>b</i>作为元素，而不包含不同于它们二者的对象<i>x</i>”。参见<a href="/wiki/%E7%A9%BA%E9%9B%86%E5%85%AC%E7%90%86" title="空集公理">空集公理</a>、<a href="/wiki/%E9%85%8D%E5%AF%B9%E5%85%AC%E7%90%86" title="配对公理">对集公理</a>。</li> <li><b>公理III</b>。<a href="/wiki/%E5%88%86%E7%B1%BB%E5%85%AC%E7%90%86" title="分类公理">分离公理</a>（Axiom der Aussonderung）：“只要<a href="/w/index.php?title=%E5%91%BD%E9%A2%98%E5%87%BD%E6%95%B0&action=edit&redlink=1" class="new" title="命题函数（页面不存在）">命题函数</a>–(<i>x</i>)对于一个集合<i>M</i>的所有元素是明确的，则存在<i>M</i>一个子集<i>M' </i>，它精确地包含<i>M</i>中使–(<i>x</i>)为真的那些元素作为元素”。</li> <li><b>公理IV</b>。<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>（Axiom der Potenzmenge）：“对于所有集合<i>T</i>都对应着一个集合<i>T' </i>，<i>T</i>的<a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">幂集</a>，精确的包含<i>T</i>的所有子集作为元素”。</li> <li><b>公理V</b>。<a href="/wiki/%E5%B9%B6%E9%9B%86%E5%85%AC%E7%90%86" title="并集公理">并集公理</a>（Axiom der Vereinigung）：“对于所有集合<i>T</i>都对应着一个集合<i>∪T</i>，<i>T</i>的并集，精确的包含<i>T</i>的元素们的所有元素作为元素”。</li> <li><b>公理VI</b>。<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>（Axiom der Auswahl）：“如果<i>T</i>是其元素都是不同于<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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span>并且相互无交的集合们的集合，它的并集<i>∪T</i>包含至少一个子集<i>S</i><sub>1</sub>有一个且只有一个元素公共于<i>T</i>的每个元素”。</li> <li><b>公理VII</b>。<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>（Axiom des Unendlichen）：“在域中存在至少一个集合<i>Z</i>包含空集作为一个元素，并且对于它的每个元素<i>a</i>都对应着形如{<i>a</i>}的进一步元素而构成的，换句话说，对于它的每个元素<i>a</i>它也包含对应的集合{<i>a</i>}作为元素”。</li></ul> <div class="mw-heading mw-heading2"><h2 id="与标准集合论的联系"><span id=".E4.B8.8E.E6.A0.87.E5.87.86.E9.9B.86.E5.90.88.E8.AE.BA.E7.9A.84.E8.81.94.E7.B3.BB"></span>与标准集合论的联系</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=2" title="编辑章节：与标准集合论的联系"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>公认的标准集合论是<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>。其中没有“基本集合公理”的完全对应者。（后来证实单元素集合可以从所谓的“对集公理”推导出来。如果<i>a</i>存在，<i>a</i>和<i>a</i>存在，所以{<i>a</i>,<i>a</i>}存在。通过外延性{<i>a</i>,<i>a</i>} = {<i>a</i>}。）空集公理已经被无穷公理所假定，现在不被包括为它的一部分了。 </p><p>这里的公理不包括<a href="/wiki/%E6%AD%A3%E8%A7%84%E5%85%AC%E7%90%86" class="mw-redirect" title="正规公理">正规公理</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>。它们是<a href="/w/index.php?title=Thoralf_Skolem&action=edit&redlink=1" class="new" title="Thoralf Skolem（页面不存在）">Thoralf Skolem</a>在1922年基于同一年早些时候<a href="/w/index.php?title=Adolf_Fraenkel&action=edit&redlink=1" class="new" title="Adolf Fraenkel（页面不存在）">Adolf Fraenkel</a>的工作而增加的。 </p><p>在现代ZFC系统中，在分离公理中提及的“命题函数”被解释为“可用带有参数的一阶公式定义的任何性质”。“一阶公式”的概念在1904年策梅洛发表他的公理的时候是未知的，而他后来拒绝这种解释因为太受限制了。 </p><p>在通常的ZFC集合论的<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">累积层次</a><i>V</i><sub>α</sub>（对于序数α）中，对于大于第一个无限序数ω的极限序数α的集合<i>V</i><sub>α</sub>之一形成了策梅洛集合论的模型。所以策梅洛集合论的相容性是ZFC集合论的一个定理。策梅洛的公理不允许很多无限基数的存在；例如，在策梅洛集合论的模型<i>V</i><sub>ω+ω</sub>中对于有限序数α只有无限基数<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 \aleph _{\alpha }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e64edf01b948311ba138926e25b7aead9ab0b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.285ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }\ }"></span>。 </p><p><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>现在通常被修改为断言第一个无限冯·诺伊曼<a href="/wiki/%E5%BA%8F%E6%95%B0" title="序数">序数</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 \omega \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/618fc4788f13fcdfe792ddf35ff04c61cfc68d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.026ex; height:1.676ex;" alt="{\displaystyle \omega \ }"></span>的存在性；有意思的是观察到最初的策梅洛公理不能证明这个集合的存在，而修改后的策梅洛公理也不能证明策梅洛的无穷公理。策梅洛的公理（最初的或修改后的）不能证明<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_{\omega }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\omega }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c94b9dcf08775bc852792575866760394b6f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.19ex; height:2.509ex;" alt="{\displaystyle V_{\omega }\ }"></span>作为一个集合的存在性，也不能证明带有无限标定（index）的累积层次的任何阶的存在性。 </p> <div class="mw-heading mw-heading2"><h2 id="策梅洛论文的目标"><span id=".E7.AD.96.E6.A2.85.E6.B4.9B.E8.AE.BA.E6.96.87.E7.9A.84.E7.9B.AE.E6.A0.87"></span>策梅洛论文的目标</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=3" title="编辑章节：策梅洛论文的目标"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>介绍声称了集合论学科的真正存在性，“它好像受到从它的原理推导出的特定矛盾或“自相矛盾”的威胁–这些原理必然支配我们的思维–而完全满意的解决似乎仍未找到。”策梅洛当然指的是<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>。 </p><p>他说希望展示<a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94" class="mw-redirect" title="康托尔">康托尔</a>和<a href="/wiki/%E6%88%B4%E5%BE%B7%E9%87%91" class="mw-redirect" title="戴德金">戴德金</a>的最初理论如何被简约到很少的定义和一些原理或公理。他说他仍未能够证明这些公理是相容的。 </p> <div class="mw-heading mw-heading2"><h2 id="分离公理"><span id=".E5.88.86.E7.A6.BB.E5.85.AC.E7.90.86"></span>分离公理</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=4" title="编辑章节：分离公理"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>策梅洛注解他的系统中的公理III负责消除悖论。它不同于康托尔最初的定义。 </p><p>集合不能用任何任意的逻辑上可定义的概念来独立的定义。它们必须被“分离”为已经“给出”的集合的子集。他说这消除了矛盾性的想法如“所有集合的集合”或“所有序数的集合”。 </p> <div class="mw-heading mw-heading2"><h2 id="康托尔定理"><span id=".E5.BA.B7.E6.89.98.E5.B0.94.E5.AE.9A.E7.90.86"></span>康托尔定理</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=5" title="编辑章节：康托尔定理"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>策梅洛的论文因第一次提及<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>而著名。它严格的凭借了集合论的概念，因此不完全同于最初的康托尔<a href="/wiki/%E5%B0%8D%E8%A7%92%E8%AB%96%E8%AD%89%E6%B3%95" title="對角論證法">对角论证法</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="引用"><span id=".E5.BC.95.E7.94.A8"></span>引用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&section=6" title="编辑章节：引用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Zermelo, Ernst. "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen, 65: 261-281, 1908. English translation, "Investigations in the foundations of set theory" in Heijenoort 1967, pages 199-215.</li></ul> <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 .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist 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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 href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">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 href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">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 class="mw-selflink selflink">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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=策梅洛集合论&oldid=65723589">https://zh.wikipedia.org/w/index.php?title=策梅洛集合论&oldid=65723589</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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