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This is a somewhat continuation of this question. The related paper is Jech's Stationary subsets of inaccessible cardinals. See also Chapter 8 and Chapter 24 of Jech's Set Theory.

I would like to ask about Theorem 3.8 of the paper. In particular, I would like to ask the last paragraph, which proves the identity (3.13). Jech wrote "...almost all $\alpha \in M_\nu$ are regular cardinals". I cannot see how this is true - if there exists a club $C$ such that all elements of $M_\nu \cap C$ are regular cardinals, then in particular it is a stationary subset of $\kappa$. This would imply that $\kappa$ is Mahlo, but no large cardinal assumptions were made in the theorem.

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  • $\begingroup$ I don't have time right now to review the relevant definitions, but I'm pretty sure that $M_\nu$ will be empty unless $\kappa$ is Mahlo and in fact Mahlo of high order ($\nu$ or thereabouts). So then "almost all $\alpha\in M_\nu$ could be no $\alpha$ at all. $\endgroup$
    – Andreas Blass
    Commented Aug 22, 2022 at 13:05
  • $\begingroup$ @AndreasBlass I don't think $M_\nu$ will always be empty. For instance, if $\kappa = \omega_2$, then $M_1 = \{\alpha < \omega_2 : \operatorname{cf}(\alpha) = \omega_1\}$ is non-empty and stationary. $M_\nu$ is indeed empty for $\nu \geq 2$, but Jech seems to claim that the proof applies for any $\nu$. $\endgroup$
    – Clement Yung
    Commented Aug 23, 2022 at 16:52
  • $\begingroup$ If you look at the paragraph before (3.13), then it seems like he is assuming that $\kappa\leq\nu<\kappa^+$ so this case would only apply in a situation where you are dealing with Mahlo cardinals. $\endgroup$
    – Todd Eisworth
    Commented Aug 25, 2022 at 0:57

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