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Available in 19 languages" > <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-19" 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">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Schr%C3%B6der-Bernstein" title="Teorema de Schröder-Bernstein – Catalan" lang="ca" hreflang="ca" data-title="Teorema de Schröder-Bernstein" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Cantorova%E2%80%93Bernsteinova_v%C4%9Bta" title="Cantorova–Bernsteinova věta – Czech" lang="cs" hreflang="cs" data-title="Cantorova–Bernsteinova věta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Cantor-Bernstein-Schr%C3%B6der" title="Satz von Cantor-Bernstein-Schröder – German" lang="de" hreflang="de" data-title="Satz von Cantor-Bernstein-Schröder" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_Cantor-Bernstein-Schr%C3%B6der" title="Teorema de Cantor-Bernstein-Schröder – Spanish" lang="es" hreflang="es" data-title="Teorema de Cantor-Bernstein-Schröder" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%DA%A9%D8%A7%D9%86%D8%AA%D9%88%D8%B1_%D8%A8%D8%B1%D9%86%D8%B4%D8%AA%D8%A7%DB%8C%D9%86" title="قضیه کانتور برنشتاین – Persian" lang="fa" hreflang="fa" data-title="قضیه کانتور برنشتاین" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Cantor-Bernstein" title="Théorème de Cantor-Bernstein – French" lang="fr" hreflang="fr" data-title="Théorème de Cantor-Bernstein" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B9%B8%ED%86%A0%EC%96%B4-%EB%B2%A0%EB%A5%B8%EC%8A%88%ED%83%80%EC%9D%B8_%EC%A0%95%EB%A6%AC" title="칸토어-베른슈타인 정리 – Korean" lang="ko" hreflang="ko" data-title="칸토어-베른슈타인 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Cantor-Schr%C3%B6der-Bernsteinov_pou%C4%8Dak" title="Cantor-Schröder-Bernsteinov poučak – Croatian" lang="hr" hreflang="hr" data-title="Cantor-Schröder-Bernsteinov poučak" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Cantor-Bernstein-Schr%C3%B6der" title="Teorema di Cantor-Bernstein-Schröder – Italian" lang="it" hreflang="it" data-title="Teorema di Cantor-Bernstein-Schröder" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A7%D7%A0%D7%98%D7%95%D7%A8-%D7%A9%D7%A8%D7%93%D7%A8-%D7%91%D7%A8%D7%A0%D7%A9%D7%98%D7%99%D7%99%D7%9F" title="משפט קנטור-שרדר-ברנשטיין – Hebrew" lang="he" hreflang="he" data-title="משפט קנטור-שרדר-ברנשטיין" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Cantor-Bernstein-Schr%C3%B6der" title="Stelling van Cantor-Bernstein-Schröder – Dutch" lang="nl" hreflang="nl" data-title="Stelling van Cantor-Bernstein-Schröder" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E3%83%B3%E3%82%B7%E3%83%A5%E3%82%BF%E3%82%A4%E3%83%B3%E3%81%AE%E5%AE%9A%E7%90%86" title="ベルンシュタインの定理 – Japanese" lang="ja" hreflang="ja" data-title="ベルンシュタインの定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Cantora-Bernsteina-Schr%C3%B6dera" title="Twierdzenie Cantora-Bernsteina-Schrödera – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie Cantora-Bernsteina-Schrödera" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Cantor-Bernstein-Schroeder" title="Teorema de Cantor-Bernstein-Schroeder – Portuguese" lang="pt" hreflang="pt" data-title="Teorema de Cantor-Bernstein-Schroeder" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</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%B5%D0%BC%D0%B0_%D0%9A%D0%B0%D0%BD%D1%82%D0%BE%D1%80%D0%B0_%E2%80%94_%D0%91%D0%B5%D1%80%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD%D0%B0" title="Теорема Кантора — Бернштейна – Russian" lang="ru" hreflang="ru" data-title="Теорема Кантора — Бернштейна" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Cantorova-Bernsteinova_veta" title="Cantorova-Bernsteinova veta – Slovak" lang="sk" hreflang="sk" data-title="Cantorova-Bernsteinova veta" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Cantorin%E2%80%93Schr%C3%B6derin%E2%80%93Bernsteinin_lause" title="Cantorin–Schröderin–Bernsteinin lause – Finnish" lang="fi" hreflang="fi" data-title="Cantorin–Schröderin–Bernsteinin lause" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</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%D0%B5%D0%BC%D0%B0_%D0%9A%D0%B0%D0%BD%D1%82%D0%BE%D1%80%D0%B0_%E2%80%94_%D0%91%D0%B5%D1%80%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD%D0%B0_%E2%80%94_%D0%A8%D1%80%D0%B5%D0%B4%D0%B5%D1%80%D0%B0" title="Теорема Кантора — Бернштейна — Шредера – Ukrainian" lang="uk" hreflang="uk" data-title="Теорема Кантора — Бернштейна — Шредера" data-language-autonym="Українська" 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vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Theorem in set theory</div> <p>In <a href="/wiki/Set_theory" title="Set theory">set theory</a>, the <b>Schröder–Bernstein theorem</b> states that, if there exist <a href="/wiki/Injective_function" title="Injective function">injective functions</a> <span class="texhtml"><i>f</i> : <i>A</i> → <i>B</i></span> and <span class="texhtml"><i>g</i> : <i>B</i> → <i>A</i></span> between the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span>, then there exists a <a href="/wiki/Bijection" title="Bijection">bijective</a> function <span class="texhtml"><i>h</i> : <i>A</i> → <i>B</i></span>. </p><p>In terms of the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the two sets, this classically implies that if <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>A</i></span>| ≤ |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>B</i></span>|</span> and <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>B</i></span>| ≤ |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>A</i></span>|</span>, then <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>A</i></span>| = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>B</i></span>|</span>; that is, <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span> are <a href="/wiki/Equipotent" class="mw-redirect" title="Equipotent">equipotent</a>. This is a useful feature in the ordering of <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>. </p><p>The theorem is named after <a href="/wiki/Felix_Bernstein_(mathematician)" title="Felix Bernstein (mathematician)">Felix Bernstein</a> and <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a>. It is also known as the <b>Cantor–Bernstein theorem</b> or <b>Cantor–Schröder–Bernstein theorem</b>, after <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, who first published it (albeit without proof). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=1" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cantor-Bernstein.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d6/Cantor-Bernstein.png" decoding="async" width="400" height="400" class="mw-file-element" data-file-width="400" data-file-height="400" /></a><figcaption>König's definition of a bijection <style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:#00c000"><i>h</i></span>:<i>A</i> → <i>B</i> from given example injections <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#c00000"><i>f</i></span>:<i>A</i> → <i>B</i> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#0000c0"><i>g</i></span>:<i>B</i> → <i>A</i>. An element in <i>A</i> and <i>B</i> is denoted by a number and a letter, respectively. The sequence 3 → e → 6 → ... is an <i>A</i>-stopper, leading to the definitions <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#00c000"><i>h</i></span>(3) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#c00000"><i>f</i></span>(3) = <i>e</i>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#00c000"><i>h</i></span>(6) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#c00000"><i>f</i></span>(6), .... The sequence <i>d</i> → 5 → <i>f</i> → ... is a <i>B</i>-stopper, leading to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#00c000"><i>h</i></span>(5) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#0000c0"><i>g</i></span><sup>−1</sup>(5) = <i>d</i>, .... The sequence ... → <i>a</i> → 1 → <i>c</i> → 4 → ... is doubly infinite, leading to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#00c000"><i>h</i></span>(1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#0000c0"><i>g</i></span><sup>−1</sup>(1) = <i>a</i>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#00c000"><i>h</i></span>(4) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#0000c0"><i>g</i></span><sup>−1</sup>(4) = <i>c</i>, .... The sequence <i>b</i> → 2 → <i>b</i> is cyclic, leading to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#00c000"><i>h</i></span>(2) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#0000c0"><i>g</i></span><sup>−1</sup>(2) = <i>b</i>.</figcaption></figure> <p>The following proof is attributed to <a href="/wiki/Julius_K%C3%B6nig" class="mw-redirect" title="Julius König">Julius König</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Assume without loss of generality that <i>A</i> and <i>B</i> are <a href="/wiki/Disjoint_set" class="mw-redirect" title="Disjoint set">disjoint</a>. For any <i>a</i> in <i>A</i> or <i>b</i> in <i>B</i> we can form a unique two-sided sequence of elements that are alternately in <i>A</i> and <i>B</i>, by repeatedly applying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> and <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 g^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae77eeb0200a3b0e26ff4b251fb845ca1b385c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:3.009ex;" alt="{\displaystyle g^{-1}}"></span> to go from <i>A</i> to <i>B</i> and <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span> to go from <i>B</i> to <i>A</i> (where defined; the inverses <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span> and <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 g^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae77eeb0200a3b0e26ff4b251fb845ca1b385c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:3.009ex;" alt="{\displaystyle g^{-1}}"></span> are understood as <a href="/wiki/Partial_function" title="Partial function">partial functions</a>.) </p> <dl><dd><i><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 \cdots \rightarrow f^{-1}(g^{-1}(a))\rightarrow g^{-1}(a)\rightarrow a\rightarrow f(a)\rightarrow g(f(a))\rightarrow \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋯</mo> <mo stretchy="false">→</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>a</mi> <mo stretchy="false">→</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdots \rightarrow f^{-1}(g^{-1}(a))\rightarrow g^{-1}(a)\rightarrow a\rightarrow f(a)\rightarrow g(f(a))\rightarrow \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65db50bdede9cd4f65acf3944ab2ee3c9e8e4b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.364ex; height:3.176ex;" alt="{\displaystyle \cdots \rightarrow f^{-1}(g^{-1}(a))\rightarrow g^{-1}(a)\rightarrow a\rightarrow f(a)\rightarrow g(f(a))\rightarrow \cdots }"></span></i></dd></dl> <p>For any particular <i>a</i>, this sequence may terminate to the left or not, at a point where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span> or <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 g^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae77eeb0200a3b0e26ff4b251fb845ca1b385c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:3.009ex;" alt="{\displaystyle g^{-1}}"></span> is not defined. </p><p>By the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> and <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> are injective functions, each <i>a</i> in <i>A</i> and <i>b</i> in <i>B</i> is in exactly one such sequence to within identity: if an element occurs in two sequences, all elements to the left and to the right must be the same in both, by the definition of the sequences. Therefore, the sequences form a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of the (disjoint) union of <i>A</i> and <i>B</i>. Hence it suffices to produce a bijection between the elements of <i>A</i> and <i>B</i> in each of the sequences separately, as follows: </p><p>Call a sequence an <i>A-stopper</i> if it stops at an element of <i>A</i>, or a <i>B-stopper</i> if it stops at an element of <i>B</i>. Otherwise, call it <i><a href="/wiki/Doubly_infinite" class="mw-redirect" title="Doubly infinite">doubly infinite</a></i> if all the elements are distinct or <i>cyclic</i> if it repeats. See the picture for examples. </p> <ul><li>For an <i>A-stopper</i>, the function <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span></i> is a bijection between its elements in <i>A</i> and its elements in <i>B</i>.</li> <li>For a <i>B-stopper</i>, the function <i><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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span></i> is a bijection between its elements in <i>B</i> and its elements in <i>A</i>.</li> <li>For a <i>doubly infinite</i> sequence or a <i>cyclic</i> sequence, either <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span></i> or <i><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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span></i> will do (<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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> is used in the picture).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Corollary_for_surjective_pair">Corollary for surjective pair</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=2" title="Edit section: Corollary for surjective pair"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we assume the axiom of choice, then a pair of surjective functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> and <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> also implies the existence of a bijection. We construct an injective function <span class="texhtml"><i>h</i> : <i>B</i> → <i>A</i></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span> by picking a single element from the inverse image of each point in <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>. The surjectivity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> guarantees the existence of at least one element in each such inverse image. We do the same to obtain an injective function <span class="texhtml"><i>k</i> : <i>A</i> → <i>B</i></span> from <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 g^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae77eeb0200a3b0e26ff4b251fb845ca1b385c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:3.009ex;" alt="{\displaystyle g^{-1}}"></span>. The Schroeder-Bernstein theorem then can be applied to the injections <i>h</i> and <i>k</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Bijective function from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]\to [0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]\to [0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a36f8747ab3ee636af08662bf42afaccc712e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.177ex; height:2.843ex;" alt="{\displaystyle [0,1]\to [0,1)}"></span></dt> <dd></dd> <dd><i>Note: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f99b30b4451167959e97802252ad13b87af5505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.843ex;" alt="{\displaystyle [0,1)}"></span> is the half open set from 0 to 1, including the boundary 0 and excluding the boundary 1.</i></dd> <dd>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:[0,1]\to [0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:[0,1]\to [0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c88768a66fc6885fdc9f4f1b5dcba0d901e0bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.393ex; height:2.843ex;" alt="{\displaystyle f:[0,1]\to [0,1)}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x/2;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x/2;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4eec5168241ea77a8fa3dd3ba1620fa5a8ea280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.817ex; height:2.843ex;" alt="{\displaystyle f(x)=x/2;}"></span> and <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 g:[0,1)\to [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:[0,1)\to [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3bc8b0eaef67dc20aeaf8bde6530b4f87ada0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.23ex; height:2.843ex;" alt="{\displaystyle g:[0,1)\to [0,1]}"></span> with <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 g(x)=x;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=x;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d705c8e12bb9d30a279895b713b1de832307ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.33ex; height:2.843ex;" alt="{\displaystyle g(x)=x;}"></span> the two injective functions.</dd> <dd>In line with that procedure <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 C_{0}=\{1\},\;C_{k}=\{2^{-k}\},\;C=\bigcup _{k=0}^{\infty }C_{k}=\{1,{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{8}},...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>C</mi> <mo>=</mo> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0}=\{1\},\;C_{k}=\{2^{-k}\},\;C=\bigcup _{k=0}^{\infty }C_{k}=\{1,{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{8}},...\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb76aa7feb11f7f8d1e7e5b3454a9c0982c3b22f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.121ex; height:7.009ex;" alt="{\displaystyle C_{0}=\{1\},\;C_{k}=\{2^{-k}\},\;C=\bigcup _{k=0}^{\infty }C_{k}=\{1,{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{8}},...\}}"></span></dd> <dd>Then <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 h(x)={\begin{cases}{\frac {x}{2}},&\mathrm {for} \ x\in C\\x,&\mathrm {for} \ x\in [0,1]\smallsetminus C\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∖</mo> <mi>C</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)={\begin{cases}{\frac {x}{2}},&\mathrm {for} \ x\in C\\x,&\mathrm {for} \ x\in [0,1]\smallsetminus C\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f46c9952a315940b96c04588d2c829039abd9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.766ex; height:6.176ex;" alt="{\displaystyle h(x)={\begin{cases}{\frac {x}{2}},&\mathrm {for} \ x\in C\\x,&\mathrm {for} \ x\in [0,1]\smallsetminus C\end{cases}}}"></span> is a bijective function from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]\to [0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]\to [0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a36f8747ab3ee636af08662bf42afaccc712e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.177ex; height:2.843ex;" alt="{\displaystyle [0,1]\to [0,1)}"></span>.</dd></dl> <dl><dt>Bijective function from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2)\to [0,1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2)\to [0,1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8a7b68262e73cc8301408d332b6caf85d6677b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.489ex; height:3.176ex;" alt="{\displaystyle [0,2)\to [0,1)^{2}}"></span></dt> <dd></dd> <dd>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:[0,2)\to [0,1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:[0,2)\to [0,1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32acd0b56955d9a889c92afecd08ba0b651ae8f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.705ex; height:3.176ex;" alt="{\displaystyle f:[0,2)\to [0,1)^{2}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(x/2;0);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(x/2;0);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704b50de684f10f8636ad9278fe952a64df0bf2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.823ex; height:2.843ex;" alt="{\displaystyle f(x)=(x/2;0);}"></span></dd> <dd>Then for <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;y)\in [0,1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x;y)\in [0,1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d54f154078d3a29fd64a87f335e249f8a6abca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.134ex; height:3.176ex;" alt="{\displaystyle (x;y)\in [0,1)^{2}}"></span> one can use the expansions <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=\sum _{k=1}^{\infty }a_{k}\cdot 10^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\sum _{k=1}^{\infty }a_{k}\cdot 10^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b579748a137bce861852d777ad2985b1c06d723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.86ex; height:6.843ex;" alt="{\displaystyle x=\sum _{k=1}^{\infty }a_{k}\cdot 10^{-k}}"></span> and <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 y=\sum _{k=1}^{\infty }b_{k}\cdot 10^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\sum _{k=1}^{\infty }b_{k}\cdot 10^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d3dd89c2b0171ddb7c6741c7bd50204d024ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.454ex; height:6.843ex;" alt="{\displaystyle y=\sum _{k=1}^{\infty }b_{k}\cdot 10^{-k}}"></span> with <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_{k},b_{k}\in \{0,1,...,9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k},b_{k}\in \{0,1,...,9\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c3ee2eec7922a27f89e93339bc4ea6ba18101b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.295ex; height:2.843ex;" alt="{\displaystyle a_{k},b_{k}\in \{0,1,...,9\}}"></span></dd> <dd>and now one can set <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 g(x;y)=\sum _{k=1}^{\infty }(10\cdot a_{k}+b_{k})\cdot 10^{-2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>10</mn> <mo>⋅</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x;y)=\sum _{k=1}^{\infty }(10\cdot a_{k}+b_{k})\cdot 10^{-2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a545a17f56925c433db608bbc6b098be82eb7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.15ex; height:6.843ex;" alt="{\displaystyle g(x;y)=\sum _{k=1}^{\infty }(10\cdot a_{k}+b_{k})\cdot 10^{-2k}}"></span> which defines an injective function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1)^{2}\to [0,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1)^{2}\to [0,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8223565433b5bb07073b191616651a6982e5a270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.489ex; height:3.176ex;" alt="{\displaystyle [0,1)^{2}\to [0,2)}"></span>. (Example: <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 g({\tfrac {1}{3}};{\tfrac {2}{3}})=0.363636...={\tfrac {4}{11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.363636...</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>11</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g({\tfrac {1}{3}};{\tfrac {2}{3}})=0.363636...={\tfrac {4}{11}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9793fc3eb43b8d5d9c26be2fbce7ceeb76d7227b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.677ex; height:3.676ex;" alt="{\displaystyle g({\tfrac {1}{3}};{\tfrac {2}{3}})=0.363636...={\tfrac {4}{11}}}"></span>)</dd> <dd>And therefore a bijective function <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> can be constructed with the use of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> and <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 g^{-1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{-1}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3308ef796aefe166c46a2e8ae57936671a91558a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.59ex; height:3.176ex;" alt="{\displaystyle g^{-1}(x)}"></span>.</dd> <dd>In this case <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 C_{0}=[1,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0}=[1,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55fea444ea6633f473a7f7afbae78497bcd7a614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.725ex; height:2.843ex;" alt="{\displaystyle C_{0}=[1,2)}"></span> is still easy but already <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 C_{1}=g(f(C_{0}))=g(\{(x;0)|x\in [{\tfrac {1}{2}},1)\,\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}=g(f(C_{0}))=g(\{(x;0)|x\in [{\tfrac {1}{2}},1)\,\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1283a8298a1555d74a5f2fa385838faad05b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.838ex; height:3.509ex;" alt="{\displaystyle C_{1}=g(f(C_{0}))=g(\{(x;0)|x\in [{\tfrac {1}{2}},1)\,\})}"></span> gets quite complicated.</dd> <dd><small>Note: Of course there's a more simple way by using the (already bijective) function definition <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 g_{2}(x;y)=2\cdot \sum _{k=1}^{\infty }(10\cdot a_{k}+b_{k})\cdot 10^{-2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>10</mn> <mo>⋅</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}(x;y)=2\cdot \sum _{k=1}^{\infty }(10\cdot a_{k}+b_{k})\cdot 10^{-2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434b41f8684cb05e6d4f10cb70de8ed75727338a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.038ex; height:6.843ex;" alt="{\displaystyle g_{2}(x;y)=2\cdot \sum _{k=1}^{\infty }(10\cdot a_{k}+b_{k})\cdot 10^{-2k}}"></span>. Then <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> would be the empty set and <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 h(x)=g_{2}^{-1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=g_{2}^{-1}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff358dfaa1752836d0aa53eb932d8f06914a1aed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.166ex; height:3.343ex;" alt="{\displaystyle h(x)=g_{2}^{-1}(x)}"></span> for all x.</small></dd></dl> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=4" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The traditional name "Schröder–Bernstein" is based on two proofs published independently in 1898. Cantor is often added because he first stated the theorem in 1887, while Schröder's name is often omitted because his proof turned out to be flawed while the name of <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>, who first proved it, is not connected with the theorem. According to Bernstein, Cantor had suggested the name <i>equivalence theorem</i> (Äquivalenzsatz).<sup id="cite_ref-Brieskorn.Chatterji.2002_2-0" class="reference"><a href="#cite_note-Brieskorn.Chatterji.2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CantorEquivalenceTheorem1887b.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/CantorEquivalenceTheorem1887b.gif/220px-CantorEquivalenceTheorem1887b.gif" decoding="async" width="220" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/CantorEquivalenceTheorem1887b.gif/330px-CantorEquivalenceTheorem1887b.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/CantorEquivalenceTheorem1887b.gif/440px-CantorEquivalenceTheorem1887b.gif 2x" data-file-width="1009" data-file-height="292" /></a><figcaption>Cantor's first statement of the theorem (1887)<sup id="cite_ref-Cantor.1932_3-0" class="reference"><a href="#cite_note-Cantor.1932-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></figcaption></figure> <ul><li><b>1887</b> <b>Cantor</b> publishes the theorem, however without proof.<sup id="cite_ref-Cantor.1932_3-1" class="reference"><a href="#cite_note-Cantor.1932-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brieskorn.Chatterji.2002_2-1" class="reference"><a href="#cite_note-Brieskorn.Chatterji.2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li><b>1887</b> On July 11, <b>Dedekind</b> proves the theorem (not relying on the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>)<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> but neither publishes his proof nor tells Cantor about it. <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a> discovered Dedekind's proof and in 1908<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> he publishes his own proof based on the <i>chain theory</i> from Dedekind's paper <i>Was sind und was sollen die Zahlen?</i><sup id="cite_ref-Brieskorn.Chatterji.2002_2-2" class="reference"><a href="#cite_note-Brieskorn.Chatterji.2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><b>1895</b> <b>Cantor</b> states the theorem in his first paper on set theory and transfinite numbers. He obtains it as an easy consequence of the linear order of cardinal numbers.<sup id="cite_ref-Cantor.1895_7-0" class="reference"><a href="#cite_note-Cantor.1895-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cantor.1895b_8-0" class="reference"><a href="#cite_note-Cantor.1895b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cantor.1897_9-0" class="reference"><a href="#cite_note-Cantor.1897-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> However, he could not prove the latter theorem, which is shown in 1915 to be equivalent to the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> by <a href="/wiki/Friedrich_Moritz_Hartogs" class="mw-redirect" title="Friedrich Moritz Hartogs">Friedrich Moritz Hartogs</a>.<sup id="cite_ref-Brieskorn.Chatterji.2002_2-3" class="reference"><a href="#cite_note-Brieskorn.Chatterji.2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li><b>1896</b> <b>Schröder</b> announces a proof (as a corollary of a theorem by <a href="/wiki/William_Stanley_Jevons" title="William Stanley Jevons">Jevons</a>).<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li><b>1897</b> <b>Bernstein</b>, a 19-year-old student in Cantor's Seminar, presents his proof.<sup id="cite_ref-Deiser.2010_12-0" class="reference"><a href="#cite_note-Deiser.2010-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Suppes.1972_13-0" class="reference"><a href="#cite_note-Suppes.1972-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li><b>1897</b> Almost simultaneously, but independently, <b>Schröder</b> finds a proof.<sup id="cite_ref-Deiser.2010_12-1" class="reference"><a href="#cite_note-Deiser.2010-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Suppes.1972_13-1" class="reference"><a href="#cite_note-Suppes.1972-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li><b>1897</b> After a visit by Bernstein, <b>Dedekind</b> independently proves the theorem a second time.</li> <li><b>1898</b> <b>Bernstein'</b>s proof (not relying on the axiom of choice) is published by <a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Émile Borel</a> in his book on functions.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> (Communicated by Cantor at the 1897 <a href="/wiki/International_Congress_of_Mathematicians" title="International Congress of Mathematicians">International Congress of Mathematicians</a> in Zürich.) In the same year, the proof also appears in <b>Bernstein'</b>s dissertation.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brieskorn.Chatterji.2002_2-4" class="reference"><a href="#cite_note-Brieskorn.Chatterji.2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li><b>1898</b> <b>Schröder</b> publishes his proof<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> which, however, is shown to be faulty by <a href="/wiki/Alwin_Reinhold_Korselt" class="mw-redirect" title="Alwin Reinhold Korselt">Alwin Reinhold Korselt</a> in 1902 (just before Schröder's death),<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> (confirmed by Schröder),<sup id="cite_ref-Brieskorn.Chatterji.2002_2-5" class="reference"><a href="#cite_note-Brieskorn.Chatterji.2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> but Korselt's paper is published only in 1911.</li></ul> <p>Both proofs of Dedekind are based on his famous 1888 memoir <i>Was sind und was sollen die Zahlen?</i> and derive it as a corollary of a proposition equivalent to statement C in Cantor's paper,<sup id="cite_ref-Cantor.1895_7-1" class="reference"><a href="#cite_note-Cantor.1895-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> which reads <span class="nowrap"><i>A</i> ⊆ <i>B</i> ⊆ <i>C</i></span> and <span class="nowrap">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>A</i></span>| = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>C</i></span>|</span> implies <span class="nowrap">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>A</i></span>| = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>B</i></span>| = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>C</i></span>|</span>. Cantor observed this property as early as 1882/83 during his studies in set theory and transfinite numbers and was therefore (implicitly) relying on the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Prerequisites">Prerequisites</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=5" title="Edit section: Prerequisites"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 1895 proof by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor</a> relied, in effect, on the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> by inferring the result as a <a href="/wiki/Corollary" title="Corollary">corollary</a> of the <a href="/wiki/Well-ordering_theorem" title="Well-ordering theorem">well-ordering theorem</a>.<sup id="cite_ref-Cantor.1895b_8-1" class="reference"><a href="#cite_note-Cantor.1895b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cantor.1897_9-1" class="reference"><a href="#cite_note-Cantor.1897-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> However, König's proof given <a href="#Proof">above</a> shows that the result can also be proved without using the axiom of choice. </p><p>On the other hand, König's proof uses the principle of <a href="/wiki/Excluded_middle" class="mw-redirect" title="Excluded middle">excluded middle</a> to draw a conclusion through case analysis. As such, the above proof is not a constructive one. In fact, in a <a href="/wiki/Constructive_set_theory" title="Constructive set theory">constructive set theory</a> such as <a href="/wiki/Intuitionism" title="Intuitionism">intuitionistic</a> set theory <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 {\mathsf {IZF}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">Z</mi> <mi mathvariant="sans-serif">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {IZF}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ebe2e4ea10a1f69c2d91c1cc67e08a7dddcccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.39ex; height:2.176ex;" alt="{\displaystyle {\mathsf {IZF}}}"></span>, which adopts the full <a href="/wiki/Axiom_of_separation" class="mw-redirect" title="Axiom of separation">axiom of separation</a> but dispenses with the principle of excluded middle, assuming the Schröder–Bernstein theorem implies the latter.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In turn, there is no proof of König's conclusion in this or weaker constructive theories. Therefore, intuitionists do not accept the statement of the Schröder–Bernstein theorem.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>There is also a proof which uses <a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Tarski's fixed point theorem</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Myhill_isomorphism_theorem" title="Myhill isomorphism theorem">Myhill isomorphism theorem</a></li> <li><a href="/wiki/Netto%27s_theorem" title="Netto's theorem">Netto's theorem</a>, according to which the bijections constructed by the Schröder–Bernstein theorem between spaces of different dimensions cannot be continuous</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem_for_measurable_spaces" title="Schröder–Bernstein theorem for measurable spaces">Schröder–Bernstein theorem for measurable spaces</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorems_for_operator_algebras" title="Schröder–Bernstein theorems for operator algebras">Schröder–Bernstein theorems for operator algebras</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_property" title="Schröder–Bernstein property">Schröder–Bernstein property</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJ._König1906" class="citation journal cs1">J. König (1906). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k30977.image.f110.langEN">"Sur la théorie des ensembles"</a>. <i>Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences</i>. <b>143</b>: 110–112.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+Rendus+Hebdomadaires+des+S%C3%A9ances+de+l%27Acad%C3%A9mie+des+Sciences&rft.atitle=Sur+la+th%C3%A9orie+des+ensembles&rft.volume=143&rft.pages=110-112&rft.date=1906&rft.au=J.+K%C3%B6nig&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k30977.image.f110.langEN&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Brieskorn.Chatterji.2002-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Brieskorn.Chatterji.2002_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Brieskorn.Chatterji.2002_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Brieskorn.Chatterji.2002_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Brieskorn.Chatterji.2002_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Brieskorn.Chatterji.2002_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Brieskorn.Chatterji.2002_2-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelix_Hausdorff2002" class="citation cs2"><a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a> (2002), <a href="/wiki/Egbert_Brieskorn" title="Egbert Brieskorn">Egbert Brieskorn</a>; Srishti D. Chatterji; et al. (eds.), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3nth_p-6DpcC"><i>Grundzüge der Mengenlehre</i></a> (1. ed.), Berlin/Heidelberg: Springer, p. 587, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-42224-2" title="Special:BookSources/978-3-540-42224-2"><bdi>978-3-540-42224-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Grundz%C3%BCge+der+Mengenlehre&rft.place=Berlin%2FHeidelberg&rft.pages=587&rft.edition=1.&rft.pub=Springer&rft.date=2002&rft.isbn=978-3-540-42224-2&rft.au=Felix+Hausdorff&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3nth_p-6DpcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span> – <a rel="nofollow" class="external text" href="https://jscholarship.library.jhu.edu/handle/1774.2/34091">Original edition (1914)</a></span> </li> <li id="cite_note-Cantor.1932-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cantor.1932_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cantor.1932_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorg_Cantor1887" class="citation cs2">Georg Cantor (1887), "Mitteilungen zur Lehre vom Transfiniten", <i><a href="/wiki/Zeitschrift_f%C3%BCr_Philosophie_und_philosophische_Kritik" title="Zeitschrift für Philosophie und philosophische Kritik">Zeitschrift für Philosophie und philosophische Kritik</a></i>, <b>91</b>: 81–125</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Philosophie+und+philosophische+Kritik&rft.atitle=Mitteilungen+zur+Lehre+vom+Transfiniten&rft.volume=91&rft.pages=81-125&rft.date=1887&rft.au=Georg+Cantor&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span><br />Reprinted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorg_Cantor1932" class="citation cs2">Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo (eds.), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN237853094&DMDID=DMDLOG_0060"><i>Gesammelte Abhandlungen mathematischen und philosophischen Inhalts</i></a>, Berlin: Springer, pp. 378–439</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gesammelte+Abhandlungen+mathematischen+und+philosophischen+Inhalts&rft.place=Berlin&rft.pages=378-439&rft.pub=Springer&rft.date=1932&rft.au=Georg+Cantor&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN237853094%26DMDID%3DDMDLOG_0060&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span> Here: p.413 bottom</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Dedekind1932" class="citation cs2">Richard Dedekind (1932), <a href="/wiki/Robert_Fricke" title="Robert Fricke">Robert Fricke</a>; Emmy Noether; Øystein Ore (eds.), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN23569441X"><i>Gesammelte mathematische Werke</i></a>, vol. 3, Braunschweig: Friedr. Vieweg & Sohn, pp. 447–449 (Ch.62)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gesammelte+mathematische+Werke&rft.place=Braunschweig&rft.pages=447-449+%28Ch.62%29&rft.pub=Friedr.+Vieweg+%26+Sohn&rft.date=1932&rft.au=Richard+Dedekind&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN23569441X&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErnst_Zermelo1908" class="citation cs2">Ernst Zermelo (1908), Felix Klein; <a href="/wiki/Walther_von_Dyck" title="Walther von Dyck">Walther von Dyck</a>; David Hilbert; <a href="/wiki/Otto_Blumenthal" title="Otto Blumenthal">Otto Blumenthal</a> (eds.), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065&DMDID=DMDLOG_0018">"Untersuchungen über die Grundlagen der Mengenlehre I"</a>, <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>65</b> (2): 261–281, here: p.271–272, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01449999">10.1007/bf01449999</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120085563">120085563</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Untersuchungen+%C3%BCber+die+Grundlagen+der+Mengenlehre+I&rft.volume=65&rft.issue=2&rft.pages=261-281%2C+here%3A+p.271-272&rft.date=1908&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120085563%23id-name%3DS2CID&rft.issn=0025-5831&rft_id=info%3Adoi%2F10.1007%2Fbf01449999&rft.au=Ernst+Zermelo&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN235181684_0065%26DMDID%3DDMDLOG_0018&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Dedekind1888" class="citation cs2">Richard Dedekind (1888), <a rel="nofollow" class="external text" href="http://echo.mpiwg-berlin.mpg.de/MPIWG:01MGQHHN"><i>Was sind und was sollen die Zahlen?</i></a> (2., unchanged (1893) ed.), Braunschweig: Friedr. Vieweg & Sohn</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Was+sind+und+was+sollen+die+Zahlen%3F&rft.place=Braunschweig&rft.edition=2.%2C+unchanged+%281893%29&rft.pub=Friedr.+Vieweg+%26+Sohn&rft.date=1888&rft.au=Richard+Dedekind&rft_id=http%3A%2F%2Fecho.mpiwg-berlin.mpg.de%2FMPIWG%3A01MGQHHN&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Cantor.1895-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cantor.1895_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cantor.1895_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorg_Cantor1932" class="citation cs2">Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo (eds.), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN237853094"><i>Gesammelte Abhandlungen mathematischen und philosophischen Inhalts</i></a>, Berlin: Springer, pp. 285 ("Satz B")</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gesammelte+Abhandlungen+mathematischen+und+philosophischen+Inhalts&rft.place=Berlin&rft.pages=285+%28%22Satz+B%22%29&rft.pub=Springer&rft.date=1932&rft.au=Georg+Cantor&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN237853094&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Cantor.1895b-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cantor.1895b_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cantor.1895b_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorg_Cantor1895" class="citation journal cs1">Georg Cantor (1895). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00225557X">"Beiträge zur Begründung der transfiniten Mengenlehre (1)"</a>. <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>. <b>46</b> (4): 481–512 (Theorem see "Satz B", p.484). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02124929">10.1007/bf02124929</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:177801164">177801164</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Beitr%C3%A4ge+zur+Begr%C3%BCndung+der+transfiniten+Mengenlehre+%281%29&rft.volume=46&rft.issue=4&rft.pages=481-512+%28Theorem+see+%22Satz+B%22%2C+p.484%29&rft.date=1895&rft_id=info%3Adoi%2F10.1007%2Fbf02124929&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A177801164%23id-name%3DS2CID&rft.au=Georg+Cantor&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fimg%2F%3FPID%3DGDZPPN00225557X&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Cantor.1897-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cantor.1897_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cantor.1897_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorg_Cantor1897" class="citation journal cs1">Georg Cantor (1897). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002256460">"Beiträge zur Begründung der transfiniten Mengenlehre (2)"</a>. <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>. <b>49</b> (2): 207–246. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01444205">10.1007/bf01444205</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121665994">121665994</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Beitr%C3%A4ge+zur+Begr%C3%BCndung+der+transfiniten+Mengenlehre+%282%29&rft.volume=49&rft.issue=2&rft.pages=207-246&rft.date=1897&rft_id=info%3Adoi%2F10.1007%2Fbf01444205&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121665994%23id-name%3DS2CID&rft.au=Georg+Cantor&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fimg%2F%3FPID%3DGDZPPN002256460&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span>)</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedrich_M._Hartogs1915" class="citation cs2">Friedrich M. Hartogs (1915), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002266105">"Über das Problem der Wohlordnung"</a>, <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>76</b> (4): 438–443, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01458215">10.1007/bf01458215</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121598654">121598654</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=%C3%9Cber+das+Problem+der+Wohlordnung&rft.volume=76&rft.issue=4&rft.pages=438-443&rft.date=1915&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121598654%23id-name%3DS2CID&rft.issn=0025-5831&rft_id=info%3Adoi%2F10.1007%2Fbf01458215&rft.au=Friedrich+M.+Hartogs&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fimg%2F%3FPID%3DGDZPPN002266105&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErnst_Schröder1896" class="citation journal cs1">Ernst Schröder (1896). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002115506">"Über G. Cantorsche Sätze"</a>. <i><a href="/wiki/Jahresbericht_der_Deutschen_Mathematiker-Vereinigung" class="mw-redirect" title="Jahresbericht der Deutschen Mathematiker-Vereinigung">Jahresbericht der Deutschen Mathematiker-Vereinigung</a></i>. <b>5</b>: 81–82.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Jahresbericht+der+Deutschen+Mathematiker-Vereinigung&rft.atitle=%C3%9Cber+G.+Cantorsche+S%C3%A4tze&rft.volume=5&rft.pages=81-82&rft.date=1896&rft.au=Ernst+Schr%C3%B6der&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fen%2Fdms%2Floader%2Fimg%2F%3FPID%3DGDZPPN002115506&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Deiser.2010-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Deiser.2010_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Deiser.2010_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOliver_Deiser2010" class="citation cs2">Oliver Deiser (2010), <i>Einführung in die Mengenlehre – Die Mengenlehre Georg Cantors und ihre Axiomatisierung durch Ernst Zermelo</i>, Springer-Lehrbuch (3rd, corrected ed.), Berlin/Heidelberg: Springer, pp. 71, 501, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-01445-1">10.1007/978-3-642-01445-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-01444-4" title="Special:BookSources/978-3-642-01444-4"><bdi>978-3-642-01444-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einf%C3%BChrung+in+die+Mengenlehre+%E2%80%93+Die+Mengenlehre+Georg+Cantors+und+ihre+Axiomatisierung+durch+Ernst+Zermelo&rft.place=Berlin%2FHeidelberg&rft.series=Springer-Lehrbuch&rft.pages=71%2C+501&rft.edition=3rd%2C+corrected&rft.pub=Springer&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-3-642-01445-1&rft.isbn=978-3-642-01444-4&rft.au=Oliver+Deiser&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Suppes.1972-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Suppes.1972_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Suppes.1972_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatrick_Suppes1972" class="citation cs2"><a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Patrick Suppes</a> (1972), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/axiomaticsettheo00supp_0/page/95"><i>Axiomatic Set Theory</i></a></span> (1. ed.), New York: Dover Publications, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/axiomaticsettheo00supp_0/page/95">95 f</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61630-8" title="Special:BookSources/978-0-486-61630-8"><bdi>978-0-486-61630-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Axiomatic+Set+Theory&rft.place=New+York&rft.pages=95+f&rft.edition=1.&rft.pub=Dover+Publications&rft.date=1972&rft.isbn=978-0-486-61630-8&rft.au=Patrick+Suppes&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faxiomaticsettheo00supp_0%2Fpage%2F95&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFÉmile_Borel1898" class="citation cs2">Émile Borel (1898), <a rel="nofollow" class="external text" href="https://archive.org/stream/leconstheoriefon00borerich#page/n115/mode/2up"><i>Leçons sur la théorie des fonctions</i></a>, Paris: Gauthier-Villars et fils, pp. 103 ff</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le%C3%A7ons+sur+la+th%C3%A9orie+des+fonctions&rft.place=Paris&rft.pages=103+ff&rft.pub=Gauthier-Villars+et+fils&rft.date=1898&rft.au=%C3%89mile+Borel&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fleconstheoriefon00borerich%23page%2Fn115%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelix_Bernstein1901" class="citation cs2">Felix Bernstein (1901), <a rel="nofollow" class="external text" href="https://archive.org/details/untersuchungena00berngoog"><i>Untersuchungen aus der Mengenlehre</i></a>, Halle a. S.: Buchdruckerei des Waisenhauses</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Untersuchungen+aus+der+Mengenlehre&rft.place=Halle+a.+S.&rft.pub=Buchdruckerei+des+Waisenhauses&rft.date=1901&rft.au=Felix+Bernstein&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funtersuchungena00berngoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span><br />Reprinted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelix_Bernstein1905" class="citation cs2">Felix Bernstein (1905), Felix Klein; Walther von Dyck; David Hilbert (eds.), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0061&DMDID=DMDLOG_0015">"Untersuchungen aus der Mengenlehre"</a>, <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>61</b> (1): 117–155, (Theorem see "Satz 1" on p.121), <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01457734">10.1007/bf01457734</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119658724">119658724</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Untersuchungen+aus+der+Mengenlehre&rft.volume=61&rft.issue=1&rft.pages=117-155%2C+%28Theorem+see+%22Satz+1%22+on+p.121%29&rft.date=1905&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119658724%23id-name%3DS2CID&rft.issn=0025-5831&rft_id=info%3Adoi%2F10.1007%2Fbf01457734&rft.au=Felix+Bernstein&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN235181684_0061%26DMDID%3DDMDLOG_0015&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErnst_Schröder1898" class="citation cs2">Ernst Schröder (1898), Kaiserliche Leopoldino-Carolinische Deutsche Akademie der Naturforscher (ed.), <a rel="nofollow" class="external text" href="https://www.biodiversitylibrary.org/item/45265#page/331/mode/1up">"Ueber zwei Definitionen der Endlichkeit und G. Cantor'sche Sätze"</a>, <i>Nova Acta</i>, <b>71</b> (6): 303–376 (proof: p.336–344)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nova+Acta&rft.atitle=Ueber+zwei+Definitionen+der+Endlichkeit+und+G.+Cantor%27sche+S%C3%A4tze&rft.volume=71&rft.issue=6&rft.pages=303-376+%28proof%3A+p.336-344%29&rft.date=1898&rft.au=Ernst+Schr%C3%B6der&rft_id=https%3A%2F%2Fwww.biodiversitylibrary.org%2Fitem%2F45265%23page%2F331%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlwin_R._Korselt1911" class="citation cs2">Alwin R. Korselt (1911), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0070&DMDID=DMDLOG_0029">"Über einen Beweis des Äquivalenzsatzes"</a>, <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>70</b> (2): 294–296, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01461161">10.1007/bf01461161</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119757900">119757900</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=%C3%9Cber+einen+Beweis+des+%C3%84quivalenzsatzes&rft.volume=70&rft.issue=2&rft.pages=294-296&rft.date=1911&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119757900%23id-name%3DS2CID&rft.issn=0025-5831&rft_id=info%3Adoi%2F10.1007%2Fbf01461161&rft.au=Alwin+R.+Korselt&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN235181684_0070%26DMDID%3DDMDLOG_0029&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Korselt (1911), p.295</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPradicBrown2019" class="citation arxiv cs1">Pradic, Cécilia; Brown, Chad E. (2019). "Cantor-Bernstein implies Excluded Middle". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1904.09193">1904.09193</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Cantor-Bernstein+implies+Excluded+Middle&rft.date=2019&rft_id=info%3Aarxiv%2F1904.09193&rft.aulast=Pradic&rft.aufirst=C%C3%A9cilia&rft.au=Brown%2C+Chad+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEttore_Carruccio2006" class="citation book cs1">Ettore Carruccio (2006). <i>Mathematics and Logic in History and in Contemporary Thought</i>. Transaction Publishers. p. 354. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-202-30850-0" title="Special:BookSources/978-0-202-30850-0"><bdi>978-0-202-30850-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Logic+in+History+and+in+Contemporary+Thought&rft.pages=354&rft.pub=Transaction+Publishers&rft.date=2006&rft.isbn=978-0-202-30850-0&rft.au=Ettore+Carruccio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Tarski's_Fixed_Point_Theorem"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Roland Uhl. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TarskisFixedPointTheorem.html">"Tarski's Fixed Point Theorem"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Tarski%27s+Fixed+Point+Theorem&rft.au=Roland+Uhl&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTarskisFixedPointTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span> Example 3.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Martin_Aigner" title="Martin Aigner">Martin Aigner</a> & <a href="/wiki/Gunter_M._Ziegler" class="mw-redirect" title="Gunter M. Ziegler">Gunter M. Ziegler</a> (1998) <a href="/wiki/Proofs_from_THE_BOOK" title="Proofs from THE BOOK">Proofs from THE BOOK</a>, § 3 Analysis: Sets and functions, <a href="/wiki/Springer_books" class="mw-redirect" title="Springer books">Springer books</a> <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1723092">1723092</a>, fifth edition 2014 <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3288091">3288091</a>, sixth edition 2018 <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3823190">3823190</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinkis2013" class="citation cs2">Hinkis, Arie (2013), <i>Proofs of the Cantor-Bernstein theorem. A mathematical excursion</i>, Science Networks. Historical Studies, vol. 45, Heidelberg: Birkhäuser/Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-0348-0224-6">10.1007/978-3-0348-0224-6</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-0348-0223-9" title="Special:BookSources/978-3-0348-0223-9"><bdi>978-3-0348-0223-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3026479">3026479</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+of+the+Cantor-Bernstein+theorem.+A+mathematical+excursion&rft.place=Heidelberg&rft.series=Science+Networks.+Historical+Studies&rft.pub=Birkh%C3%A4user%2FSpringer&rft.date=2013&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3026479%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-0348-0224-6&rft.isbn=978-3-0348-0223-9&rft.aulast=Hinkis&rft.aufirst=Arie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSearcóid2013" class="citation journal cs1">Searcóid, Míchaél Ó (2013). "On the history and mathematics of the equivalence theorem". <i><a href="/wiki/Mathematical_Proceedings_of_the_Royal_Irish_Academy" class="mw-redirect" title="Mathematical Proceedings of the Royal Irish Academy">Mathematical Proceedings of the Royal Irish Academy</a></i>. <b>113A</b> (2): 151–68. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1353%2Fmpr.2013.0006">10.1353/mpr.2013.0006</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/42912521">42912521</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:245841055">245841055</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Royal+Irish+Academy&rft.atitle=On+the+history+and+mathematics+of+the+equivalence+theorem&rft.volume=113A&rft.issue=2&rft.pages=151-68&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A245841055%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F42912521%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1353%2Fmpr.2013.0006&rft.aulast=Searc%C3%B3id&rft.aufirst=M%C3%ADcha%C3%A9l+%C3%93&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6der%E2%80%93Bernstein_theorem&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Schröder-Bernstein_Theorem"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Schroeder-BernsteinTheorem.html">"Schröder-Bernstein Theorem"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Schr%C3%B6der-Bernstein+Theorem&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSchroeder-BernsteinTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchr%C3%B6der%E2%80%93Bernstein+theorem" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/Cantor-Schroeder-Bernstein+theorem">Cantor-Schroeder-Bernstein theorem</a> at the <a href="/wiki/NLab" title="NLab"><i>n</i>Lab</a></li> <li><a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/10.1007%2Fs00283-011-9242-3.pdf">Cantor-Bernstein’s Theorem in a Semiring</a> by Marcel Crabbé.</li> <li><i>This article incorporates material from the <a href="/wiki/Citizendium" title="Citizendium">Citizendium</a> article "<a href="https://en.citizendium.org/wiki/Schr%C3%B6der-Bernstein_theorem" class="extiw" title="citizendium:Schröder-Bernstein theorem">Schröder-Bernstein_theorem</a>", which is licensed under the <a href="/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_License" class="mw-redirect" title="Wikipedia:Text of Creative Commons Attribution-ShareAlike 3.0 Unported License">Creative Commons Attribution-ShareAlike 3.0 Unported License</a> but not under the <a href="/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License" title="Wikipedia:Text of the GNU Free Documentation License">GFDL</a>.</i></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Venn_diagram" title="Venn diagram"><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/Axiom" title="Axiom">Axioms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li> <li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">Choice</a> <ul><li><a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li> <li><a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li> <li><a href="/wiki/Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a> <ul><li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">projective</a></li></ul></li> <li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li> <li><a href="/wiki/Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Union</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li></ul> <ul><li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> <ul><li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li> <li><a href="/wiki/Power_set" title="Power set">Power set</a></li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost" title="Almost">Almost</a></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="/wiki/Large_cardinal" title="Large cardinal">large</a>)</li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li><a href="/wiki/Constructible_universe" title="Constructible universe">Constructible universe</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li> <li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a> <ul><li><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></li> <li><a href="/wiki/Tuple" title="Tuple">tuple</a></li></ul></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Bijection" title="Bijection">One-to-one correspondence</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li> <li><a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_set" title="Amorphous set">Amorphous</a></li> <li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a> (<a href="/wiki/Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li> <li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">Filter</a> <ul><li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">base</a></li> <li><a href="/wiki/Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li> <li><a href="/wiki/Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a> (<a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li> <li><a href="/wiki/Computable_set" title="Computable set">Recursive</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Subset" title="Subset">Subset <b>·</b> Superset</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li> <li><a href="/wiki/Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul> <ul><li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo</a> <ul><li><a href="/wiki/General_set_theory" title="General set theory">General</a></li></ul></li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> <ul><li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li></ul></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a> <ul><li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Set_theorists" title="Category:Set theorists">Set theorists</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li> <li><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a class="mw-selflink selflink">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of 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