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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Beth-Funktion</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Beth-Funktion</b>, benannt nach dem <a href="Beth" title="Beth">zweiten Buchstaben</a> des <a href="Hebr%C3%A4isches_Alphabet" title="Hebräisches Alphabet">hebräischen Alphabets</a> und auch als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ℶ<!-- ℶ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6ca52be7fa2a32db700e775beb85f6dec33cc4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.05ex; width:1.647ex; height:2.343ex;" alt="{\displaystyle \beth }" loading="lazy"></span> geschrieben, ist eine in der <a href="Mengenlehre" title="Mengenlehre">Mengenlehre</a>, genauer in der Theorie der <a href="Kardinalzahl_(Mathematik)" title="Kardinalzahl (Mathematik)">Kardinalzahlen</a>, verwendete Aufzählung gewisser unendlicher Kardinalzahlen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Die Beth-Funktion ordnet jeder <a href="Ordinalzahl" title="Ordinalzahl">Ordinalzahl</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> eine wie folgt rekursiv definierte Kardinalzahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/602728797c3415a4581fc2ced51ea8c91692722d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.05ex; width:2.884ex; height:2.676ex;" alt="{\displaystyle \beth _{\alpha }}" loading="lazy"></span> zu:<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>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{0}=\aleph _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth _{0}=\aleph _{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e9fd73034225e9d199f58b59a0c9fa12576df0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.05ex; width:8.227ex; height:2.676ex;" alt="{\displaystyle \beth _{0}=\aleph _{0}}" loading="lazy"></span>, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}" loading="lazy"></span> die kleinste unendliche Kardinalzahl ist, siehe <a href="Aleph-Funktion" title="Aleph-Funktion">Aleph-Funktion</a>.</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64d74b311ba5b42d7e060f312aa488eb0f571c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.05ex; width:11.593ex; height:3.176ex;" alt="{\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }}}" loading="lazy"></span> für <a href="Nachfolger-Ordinalzahl" class="mw-redirect" title="Nachfolger-Ordinalzahl">Nachfolger-Ordinalzahlen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha +1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8de281682e16a35de79915ddc9e3582d215e0ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle \alpha +1}" loading="lazy"></span>. Dabei steht die rechte Seite für die <a href="Kardinalzahlarithmetik" title="Kardinalzahlarithmetik">Potenz von Kardinalzahlen</a>.</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{\lambda }=\sup _{\alpha <\lambda }\beth _{\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>λ<!-- λ --></mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">sup</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo><</mo>
<mi>λ<!-- λ --></mi>
</mrow>
</munder>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth _{\lambda }=\sup _{\alpha <\lambda }\beth _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ab2cd9535f29e3fc78bdcb0574e3aad7313c780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.05ex; width:12.612ex; height:4.509ex;" alt="{\displaystyle \beth _{\lambda }=\sup _{\alpha <\lambda }\beth _{\alpha }}" loading="lazy"></span> für <a href="Limes-Ordinalzahl" class="mw-redirect" title="Limes-Ordinalzahl">Limes-Ordinalzahlen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Bemerkungen">Bemerkungen</h2></div>
<p>Die <a href="Kontinuumshypothese" title="Kontinuumshypothese">Kontinuumshypothese</a> ist gleichbedeutend mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}=\beth _{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}=\beth _{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0a255e148da8fcda691efd9d58be423d676ad93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.178ex; height:2.676ex;" alt="{\displaystyle \aleph _{1}=\beth _{1}}" loading="lazy"></span>, denn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth _{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bb3462a86543187911778e6ff64ed1dc27b19f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.05ex; width:2.655ex; height:2.676ex;" alt="{\displaystyle \beth _{1}}" loading="lazy"></span> ist definitionsgemäß die Mächtigkeit der <a href="Potenzmenge" title="Potenzmenge">Potenzmenge</a> einer <a href="Abz%C3%A4hlbarkeit" class="mw-redirect" title="Abzählbarkeit">abzählbaren</a> Menge und daher gleichmächtig zum Kontinuum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>. Die <a href="Verallgemeinerte_Kontinuumshypothese" class="mw-redirect" title="Verallgemeinerte Kontinuumshypothese">verallgemeinerte Kontinuumshypothese</a> ist äquivalent zu <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph =\beth }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mo>=</mo>
<mi>ℶ<!-- ℶ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph =\beth }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1cee5b07ff082cf6dfe78b170080f131f4e3d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.116ex; height:2.343ex;" alt="{\displaystyle \aleph =\beth }" loading="lazy"></span>, das heißt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{\alpha }=\beth _{\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }=\beth _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1abd792cca5dba4e5a0f5e2548769c33369b1993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.637ex; height:2.676ex;" alt="{\displaystyle \aleph _{\alpha }=\beth _{\alpha }}" loading="lazy"></span> für alle Ordinalzahlen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>.
</p><p>Eine <a href="Limes-Kardinalzahl" class="mw-redirect" title="Limes-Kardinalzahl">Limes-Kardinalzahl</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>κ<!-- κ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"></span> heißt ein starker Limes, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{\lambda }<\kappa }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>λ<!-- λ --></mi>
</mrow>
</msup>
<mo><</mo>
<mi>κ<!-- κ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{\lambda }<\kappa }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72aa16b0e597dec2a2e92bea2e6dc72149ff6b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.03ex; height:3.176ex;" alt="{\displaystyle \mu ^{\lambda }<\kappa }" loading="lazy"></span> für alle Kardinalzahlen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ,\mu <\kappa }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mo>,</mo>
<mi>μ<!-- μ --></mi>
<mo><</mo>
<mi>κ<!-- κ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda ,\mu <\kappa }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eea95b86d88d2e90eba8c5247fba8cc26d42d7e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.228ex; height:2.676ex;" alt="{\displaystyle \lambda ,\mu <\kappa }" loading="lazy"></span>. Eine Kardinalzahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>κ<!-- κ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"></span> ist genau dann eine starke Limes-Kardinalzahl, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa =\beth _{\xi }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>κ<!-- κ --></mi>
<mo>=</mo>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ξ<!-- ξ --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \kappa =\beth _{\xi }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab6522865ba07e5a488c844a39bdb3dd5fa8b25e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.948ex; height:3.009ex;" alt="{\displaystyle \kappa =\beth _{\xi }}" loading="lazy"></span> für eine Limes-Ordinalzahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ξ<!-- ξ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"></span> ist.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Es gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \leq \aleph _{\alpha }\leq \beth _{\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha \leq \aleph _{\alpha }\leq \beth _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddc090826edc7115c5d587b5131d7f467142cdf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.223ex; height:2.676ex;" alt="{\displaystyle \alpha \leq \aleph _{\alpha }\leq \beth _{\alpha }}" loading="lazy"></span> für alle Ordinalzahlen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>. Man kann zeigen, dass es <a href="Fixpunkt_(Mathematik)" title="Fixpunkt (Mathematik)">Fixpunkte</a> geben muss, das heißt solche Ordinalzahlen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, für die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\beth _{\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>=</mo>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha =\beth _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cdad2a2bd2acbdd9db36061e0f6aa1faf98cec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.421ex; height:2.676ex;" alt="{\displaystyle \alpha =\beth _{\alpha }}" loading="lazy"></span> gilt. Der kleinste Fixpunkt ist der Limes der Folge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{0},\beth _{\beth _{0}},\beth _{\beth _{\beth _{0}}},\ldots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth _{0},\beth _{\beth _{0}},\beth _{\beth _{\beth _{0}}},\ldots }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/399bfb61f4abd4ac92f279bc24e046644799fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; margin-left: -0.05ex; width:16.924ex; height:3.676ex;" alt="{\displaystyle \beth _{0},\beth _{\beth _{0}},\beth _{\beth _{\beth _{0}}},\ldots }" loading="lazy"></span>, der informal als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{\beth _{{}_{\ddots }}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ℶ<!-- ℶ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>⋱<!-- ⋱ --></mo>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beth _{\beth _{{}_{\ddots }}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1accb1a2a21a78b3455b526854a3f0b1802b5ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.138ex; margin-left: -0.05ex; margin-bottom: -0.367ex; width:4.936ex; height:4.509ex;" alt="{\displaystyle \beth _{\beth _{{}_{\ddots }}}}" loading="lazy"></span> dargestellt wird. Ebenso sind <a href="Stark_unerreichbare_Kardinalzahl" class="mw-redirect" title="Stark unerreichbare Kardinalzahl">stark unerreichbare Kardinalzahlen</a> Fixpunkte der Beth-Funktion.
</p>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="Thomas_Jech" title="Thomas Jech">Thomas Jech</a>: <i>Set Theory.</i> 3rd millennium edition, revised and expanded. Springer, Berlin u. a. 2003, ISBN 3-540-44085-2, Kapitel I.5, S. 55.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">W. Wistar Comfort, Stylianos Negrepontis: <i>The Theory of Ultrafilters</i> (= <i><a href="Grundlehren_der_mathematischen_Wissenschaften" title="Grundlehren der mathematischen Wissenschaften">Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen</a>.</i> Bd. 211). Springer, Berlin u. a. 1974, ISBN 3-540-06604-7, Lemma 1.23.</span>
</li>
</ol></div><!--htdig_noindex--><div><div class="zim-footer">
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