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


  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.
  • $\begingroup$ 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 $V[C]$? $\endgroup$ 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


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.

  • $\begingroup$ 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). $\endgroup$ Jul 30, 2015 at 14:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.