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An inaccessible cardinal (or strongly inaccessible cardinal) is a cardinal number that is an uncountable regular strong limit cardinal. The smallest inaccessible cardinal is sometimes called the inaccessible cardinal \(I\). Breaking down the definition, an inaccessible cardinal \(\alpha\) must be:

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  • Inaccessible cardinal
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  • An inaccessible cardinal (or strongly inaccessible cardinal) is a cardinal number that is an uncountable regular strong limit cardinal. The smallest inaccessible cardinal is sometimes called the inaccessible cardinal \(I\). Breaking down the definition, an inaccessible cardinal \(\alpha\) must be:
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abstract
  • An inaccessible cardinal (or strongly inaccessible cardinal) is a cardinal number that is an uncountable regular strong limit cardinal. The smallest inaccessible cardinal is sometimes called the inaccessible cardinal \(I\). Breaking down the definition, an inaccessible cardinal \(\alpha\) must be: * Uncountable: \(\alpha \geq \omega_1\). * Regular: \(\alpha\) cannot be expressed as the limit of a set \(S\) of smaller ordinals, where the order type of \(S\) is less than \(\alpha\). From a cardinal perspective, we can informally say that it cannot be divided into smaller set of smaller sets. * Strong limit: \(\alpha = \beth_\gamma\) for a limit ordinal \(\gamma\), using the following hierarchy of beth numbers: * \(\beth_0 = \aleph_0\) * \(\beth_{\alpha + 1} = 2^{\beth_\alpha}\) (cardinal exponentiation) * \(\beth_\alpha = \sup\{\beta < \alpha : \beth_\beta\}\) If we replace "strong limit cardinal" with "limit cardinal" (replacing "beth numbers" with "aleph numbers"), we get weakly inaccessible cardinals. The distinction between strongly and weakly inaccessible cardinals only matters if we don't assume generalized continuum hypothesis (GCH). Under GCH, all limit cardinals are strong limit cardinals.
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