From 09ca2d59572cf839834499c4578cce8b8af4c7de Mon Sep 17 00:00:00 2001 From: zeramorphic Date: Mon, 27 May 2024 11:23:36 +0100 Subject: [PATCH] Another typo Signed-off-by: zeramorphic --- iii/lc/01_inaccessible_cardinals.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/iii/lc/01_inaccessible_cardinals.tex b/iii/lc/01_inaccessible_cardinals.tex index 761c38d..1d7dc46 100644 --- a/iii/lc/01_inaccessible_cardinals.tex +++ b/iii/lc/01_inaccessible_cardinals.tex @@ -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.