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Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo?

Remarks:

  1. It is possible for every stationary subset of $\kappa$ to reflect, but $\kappa$ is only weakly inaccessible (and not strongly inaccessible).
  2. If $V=L$ then the answer is "yes", and in fact $\kappa$ must be weakly compact.
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  • $\begingroup$ In the case of Remark 1, is $\kappa$ at least weakly Mahlo? $\endgroup$
    – Asaf Karagila ♦
    Jul 29, 2015 at 22:17
  • $\begingroup$ In the model I know of, yes. $\endgroup$
    – Sean Cox
    Jul 29, 2015 at 22:21
  • $\begingroup$ Do you know what happens if you simply use the Mahlo-killing forcing? (Conditions are closed bounded sets containing no regular cardinal.) This forcing is very nice, and has $\delta$-closed dense subsets for every $\delta<\kappa$; so it adds no bounded sets. If you have stationary reflection in the ground model, is this preserved to the forcing extension $V[C]$? $\endgroup$
    – Joel David Hamkins
    Jul 30, 2015 at 0:07
  • $\begingroup$ Joel, I suspect something like thus might work, but only if you do some sort of Prikry preparation. So that $\kappa $ is singular on the $ j $ side of the forcing, where $ j $ is a generic lifting of a ground model elementary embedding. But even then it's not clear. $\endgroup$
    – Sean Cox
    Jul 30, 2015 at 2:05

1 Answer 1

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Not necessarily. Here's a counterexample, but I'm sure it is a ridiculous overkill in consistency strength:

Suppose $\kappa$ is the least inaccessible limit of supercompact cardinals. Then $\kappa$ is not Mahlo. If $S \subseteq \kappa$ is stationary, then there is a stationary $T \subseteq S$ such that $T$ concentrates on some cofinality $\theta < \kappa$. Taking a supercompact $\delta \in (\theta,\kappa)$ gives the reflection of $T$ by the usual argument.

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  • $\begingroup$ That's fine. I didn't care about the consistency strength. $\endgroup$
    – Sean Cox
    Jul 30, 2015 at 1:59
  • $\begingroup$ The model in Shelah's paper "Reflecting stationary sets and successors of singular cardinals" has the property that stationary reflection holds for ALL inaccessible non-Mahlo cardinals (and such cardinals exist there). $\endgroup$
    – Todd Eisworth
    Jul 30, 2015 at 14:53

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