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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Aleph-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>Aleph-Funktion</b>, benannt nach dem <a href="Aleph" title="Aleph">ersten 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 \aleph }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/306c55e6bc96d94db729ff5821c8f45a34c72bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }" 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 aller unendlichen Kardinalzahlen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Die Klasse der unendlichen Kardinalzahlen ist unter Verwendung des <a href="Auswahlaxiom" title="Auswahlaxiom">Auswahlaxioms</a> in der Klasse <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 \mathbf {On} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">O</mi>
<mi mathvariant="bold">n</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {On} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eebff303626a5ca6b29cf76a9479ad24883abeac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.493ex; height:2.176ex;" alt="{\displaystyle \mathbf {On} }" loading="lazy"></span> der <a href="Ordinalzahl" title="Ordinalzahl">Ordinalzahlen</a> enthalten, wobei jede 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> mit der kleinsten 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 \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> gleichmächtigen Ordinalzahl identifiziert wird. Ferner ist das <a href="Infimum_und_Supremum" title="Infimum und Supremum">Supremum</a> einer Menge von Kardinalzahlen stets wieder eine Kardinalzahl. Daher gibt es genau einen <a href="Ordnungsisomorphismus" title="Ordnungsisomorphismus">Ordnungsisomorphismus</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 \aleph }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/306c55e6bc96d94db729ff5821c8f45a34c72bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }" loading="lazy"></span> von <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 \mathbf {On} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">O</mi>
<mi mathvariant="bold">n</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {On} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eebff303626a5ca6b29cf76a9479ad24883abeac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.493ex; height:2.176ex;" alt="{\displaystyle \mathbf {On} }" loading="lazy"></span> auf die Klasse der unendlichen Kardinalzahlen. Den Wert von <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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/306c55e6bc96d94db729ff5821c8f45a34c72bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }" loading="lazy"></span> an der Stelle <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> bezeichnet man 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 _{\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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}" 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 }}">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}" loading="lazy"></span> ist 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 }">
<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>-te unendliche Kardinalzahl.
</p><p>Die Aleph-Funktion lässt sich mit <a href="Transfinite_Rekursion" class="mw-redirect" title="Transfinite Rekursion">transfiniter Rekursion</a> wie folgt definieren:
</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 \aleph _{0}=\left|\omega \right|}">
<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>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mi>ω<!-- ω --></mi>
<mo>|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{0}=\left|\omega \right|}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/308f49ca51700eff16dd1240841856a064d9e0ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.312ex; height:2.843ex;" alt="{\displaystyle \aleph _{0}=\left|\omega \right|}" loading="lazy"></span> ist kleinste unendliche Ordinalzahl und damit auch kleinste unendliche Kardinalzahl,</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 \aleph _{\alpha +1}=\min _{\kappa >\aleph _{\alpha }}\kappa }">
<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>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">min</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>κ<!-- κ --></mi>
<mo>></mo>
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mrow>
</munder>
<mi>κ<!-- κ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha +1}=\min _{\kappa >\aleph _{\alpha }}\kappa }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dec1a243fcb7bd557852f8b51f122c993a0100ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.877ex; height:4.176ex;" alt="{\displaystyle \aleph _{\alpha +1}=\min _{\kappa >\aleph _{\alpha }}\kappa }" loading="lazy"></span>, also die kleinste Kardinalzahl, die größer 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 \aleph _{\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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}" loading="lazy"></span> ist,</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 \aleph _{\lambda }=\sup _{\alpha <\lambda }\aleph _{\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>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 mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\lambda }=\sup _{\alpha <\lambda }\aleph _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cab3c547342b0392e494e97d3858b57abff66ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.302ex; height:4.343ex;" alt="{\displaystyle \aleph _{\lambda }=\sup _{\alpha <\lambda }\aleph _{\alpha }}" loading="lazy"></span> für Limes-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 \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="Eigenschaften">Eigenschaften</h2></div>
<p>Die kleinste unendliche Kardinalzahl ist <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 Kardinalität der <a href="Abz%C3%A4hlbare_Menge" title="Abzählbare Menge">abzählbar unendlichen</a> Mengen. Die Nachfolger-Kardinalzahl, das heißt die kleinste Kardinalzahl größer 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 \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>, ist <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}}">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed.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 _{1}}" loading="lazy"></span>, und so weiter. Die Frage, ob <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}}">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed.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 _{1}}" loading="lazy"></span> gleich der Mächtigkeit der Menge der reellen Zahlen ist, ist als <a href="Kontinuumshypothese" title="Kontinuumshypothese">Kontinuumshypothese</a> bekannt.
</p><p>Allgemein ist <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 }}">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}" loading="lazy"></span> eine <i>Nachfolger-Kardinalzahl,</i> falls <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 <a href="Ordinalzahl" title="Ordinalzahl">Nachfolger-Ordinalzahl</a> ist, anderenfalls eine <i>Limes-Kardinalzahl.</i>
</p><p>Üblicherweise bezeichnet <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 \omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> die kleinste unendliche Ordinalzahl. Diese ist gleich <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>, aber als Index für die Aleph-Funktion verwendet man lieber die Ordinalzahl-Schreibweise. <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 _{\omega }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- ω --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\omega }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fab056732a610ab69363c5e2bd54dfe0217d1883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.675ex; height:2.509ex;" alt="{\displaystyle \aleph _{\omega }}" loading="lazy"></span> ist damit die kleinste Limes-Kardinalzahl und kann 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 \sup _{n<\omega }\aleph _{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">sup</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo><</mo>
<mi>ω<!-- ω --></mi>
</mrow>
</munder>
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sup _{n<\omega }\aleph _{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75d728e44def23a26a80311990e86c672976ac20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.527ex; height:4.176ex;" alt="{\displaystyle \sup _{n<\omega }\aleph _{n}}" loading="lazy"></span> geschrieben werden.
</p><p>Es gilt stets <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 }}">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha \leq \aleph _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf5bfce1d40e83e745f506e96a11e28b739feaf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.291ex; height:2.509ex;" alt="{\displaystyle \alpha \leq \aleph _{\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 =\aleph _{\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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha =\aleph _{\alpha }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dd71c3f46a4db9ffad35051299d32ff322aa0e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.291ex; height:2.509ex;" alt="{\displaystyle \alpha =\aleph _{\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 \aleph _{0},\aleph _{\aleph _{0}},\aleph _{\aleph _{\aleph _{0}}},\ldots }">
<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>
<mo>,</mo>
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></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 \aleph _{0},\aleph _{\aleph _{0}},\aleph _{\aleph _{\aleph _{0}}},\ldots }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a049e4879f779d94195bd165f63438c9655f8085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:16.225ex; height:3.343ex;" alt="{\displaystyle \aleph _{0},\aleph _{\aleph _{0}},\aleph _{\aleph _{\aleph _{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 \aleph _{\aleph _{\ddots }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⋱<!-- ⋱ --></mo>
</mrow>
</msub>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \aleph _{\aleph _{\ddots }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6520d427872e291132090b11d773efeeda4ac80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.679ex; margin-bottom: -0.326ex; width:4.531ex; height:3.843ex;" alt="{\displaystyle \aleph _{\aleph _{\ddots }}}" loading="lazy"></span> dargestellt wird. Ebenso sind <a href="Schwach_unerreichbare_Kardinalzahl" class="mw-redirect" title="Schwach unerreichbare Kardinalzahl">schwach unerreichbare Kardinalzahlen</a> Fixpunkte der Aleph-Funktion.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Anfangszahl" title="Anfangszahl">Anfangszahl</a></li>
<li><a href="Beth-Funktion" title="Beth-Funktion">Beth-Funktion</a></li>
<li><a href="Gimel-Funktion" class="mw-redirect" title="Gimel-Funktion">Gimel-Funktion</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a>: <i>Über unendliche, lineare Punktmannigfaltigkeiten. Arbeiten zur Mengenlehre aus den Jahren 1872–1884</i> (= <i>Teubner-Archiv zur Mathematik.</i> Bd. 2, <span class="-print"><a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220233-0962%22&key=cql">0233-0962</a></span></span>). Herausgegeben und kommentiert von <a href="G%C3%BCnter_Asser" title="Günter Asser">G. Asser</a>. Teubner, Leipzig, 1884.</li>
<li><a href="Thomas_Jech" title="Thomas Jech">Thomas Jech</a>: <i>Set Theory</i>. The Third Millennium Edition, revised and expanded. Springer, Berlin u. a. 2003, ISBN 3-540-44085-2.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2023-10-27" href="https://de.wikipedia.org/wiki/?title=Aleph-Funktion&oldid=238547091">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
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