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Another typo
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Signed-off-by: zeramorphic <[email protected]>
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zeramorphic committed May 27, 2024
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Expand Up @@ -379,7 +379,7 @@ \subsection{The consistency strength hierarchy}
Thus, if \( \mathsf{ZFC} + \neg\mathsf{IC} \) is consistent,
\[ \mathsf{ZFC} + \neg\mathsf{IC} <_{\Con} \mathsf{ZFC} + \mathsf{IC} \]
Observe that none of the proofs given in this section work for weakly inaccessible cardinals, so it is not clear that weakly inaccessible cardinals qualify as large cardinals.
However, that under the generalised continuum hypothesis, we have \( \aleph_\alpha = \beth_\alpha \) and so the notions of weakly inaccessible cardinal and inaccessible cardinal coincide.
However, under the generalised continuum hypothesis, we have \( \aleph_\alpha = \beth_\alpha \) and so the notions of weakly inaccessible cardinal and inaccessible cardinal coincide.
In Part III Forcing and the Continuum Hypothesis, we see that if \( M \vDash \mathsf{ZFC} \), there is \( \mathrm{L} \subseteq M \) such that \( \mathrm{L} \) is transitive in \( M \), \( \mathrm{L} \) contains all the ordinals of \( M \), and \( \mathrm{L} \vDash \mathsf{ZFC} + \mathsf{GCH} \).
Thus, given a model \( M \vDash \mathsf{ZFC} + \mathsf{WIC} \), we obtain \( \mathrm{L} \vDash \mathsf{ZFC} + \mathsf{IC} \), and thus the two axioms \( \mathsf{WIC} \) and \( \mathsf{IC} \) are equiconsistent.

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