# Does stationary reflection imply Mahloness?

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.
• In the case of Remark 1, is $\kappa$ at least weakly Mahlo? Jul 29, 2015 at 22:17
• In the model I know of, yes. Jul 29, 2015 at 22:21
• 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 $V[C]$? Jul 30, 2015 at 0:07
• 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. Jul 30, 2015 at 2:05

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.