Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal

Question

Is anything known about the consistency strength of the following statement?

• $\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap \kappa$ is an inaccessible cardinal and the order type of $a$ is $(a \cap \kappa)^+$?

This statement follows from $\kappa^+$-supercompactness of $\kappa$ and also from subcompactness of $\kappa$. It is a strengthening of the principle $\text{Pr}(\kappa^+)$ that was shown by Burke in "Generic embeddings and the failure of box" to imply $\neg \square_\kappa$, so a lower bound on the consistency strength is the existence of two Mahlo cardinals.

Answer 1

Well, here is a very slight weakening of your $\kappa^+$-supercompactness upper bound, to the assumption merely that $\kappa$ is nearly $\kappa^+$-supercompact. This hypothesis is strictly weaker than $\kappa^+$-supercompactness, but still, under this assumption, the set of such $a$ is stationary as you desire.

Specifically, a cardinal $\kappa$ is nearly $\kappa^+$-supercompact, if for every transitive $M$ of size $\kappa^+$, closed under $\lt\kappa$ sequences, there is an elementary embedding $j:M\to N$ into a transitive set $N$, with critical point $\kappa$, such that $j''\kappa^+\in N$. It follows that $j''\kappa^+$ has your property with respect to $j(\kappa)$ in $N$, and so the set of $a\in P_\kappa(\kappa^+)$ with your property will have measure one with respect to the induced $M$-measure. It follows by the usual normality argument that the set of such $a$ is stationary, as you desire, since any club set $C$ (or the function defining it) can be placed into such an $M$ and so $j''\kappa^+\in j(C)$, meaning that $j$ of the set of $a$ meets $j(C)$, and so there is such an $a$ in $C$.

The nearly $\theta$-supercompact cardinals were introduced by Jason Schanker in his dissertation (written under my supervision), along with the weakly measurable cardinals, and Jason had noted that the near $\theta$-supercompactness hypothesis suffice in many arguments that previously had used full $\theta$-supercompactness. The present case is an additional instance of this phenomenon, where near $\kappa^+$-supercompactness suffices in place of $\kappa^+$-supercompactness.

Meanwhile, I am unsure exactly how the nearly $\kappa^+$-supercompact cardinals relate to the subcompact cardinals, and in any case perhaps a more dramatic weakening is possible.

It may be interesting to note that a recent result proved jointly by Jason Schanker, Brent Cody, Moti Gitik and myself shows that it is relatively consistent with ZFC that the least weakly compact cardinal $\kappa$ is also nearly $\kappa^+$-supercompact (but $2^\kappa\gt\kappa^+$), which might suggest something about the nature of these cardinals.

Answer 2

I hope revisiting an old question is not frowned upon, but i think i have been able to improve both the lower and the upper bound of the consistency strength:

For the lower bound, Lemma 38.11 in Jech states that (for $\kappa$ regular and $\lambda\geq\kappa$) "If $\{x\in P_{\kappa}(\lambda)\;|\;|x\cap\kappa|<|x|\}$ is stationary then $0^{\#}$ exists". The proof is not too hard: By assumption, there exists $M\in P_{\kappa}(L_{\lambda})$ such that $M\prec L_{\lambda}$ and $|M\cap\kappa|<|M|$. Ergo the inverse of the transitive collapse of $M$ is an elementary embedding from some $L_{\alpha}$ (where necessarily $|M|=|L_{\alpha}|=|\alpha|>M\cap\kappa$) to $L_{\lambda}$. So by Theorem 18.27 in Jech, $0^{\#}$ exists.

On the other hand, for the lower bound the $\kappa^+$-ineffability of $\kappa$ suffices. $\kappa$ is $\kappa^+$-ineffable if for every function $f\colon P_{\kappa}(\kappa^+)\to P_{\kappa}(\kappa^+)$ such that $f(x)\subseteq x$ for all $x$, there exists $b\subseteq\kappa^+$ such that $\{x\in P_{\kappa}(\kappa^+)\;|\;f(x)=x\cap b\}$ is stationary. If $\kappa$ is nearly $\kappa^+$-supercompact, it is $\kappa^+$-ineffable (and my best guess is that the two properties are not equivalent).

In my thesis (forthcoming) i show that if $\kappa$ is $\kappa^+$-ineffable there are stationarily many $M\prec H(\Theta)$ (for $\Theta$ arbitrary) of size ${<}\,\kappa$ such that $M\cap\kappa$ is inaccessible and $M$ is "$\Pi_1^1$-correct about $\kappa^+$", meaning that (letting $\pi\colon M\to N$ denote the transitive collapse) if for some $B\subseteq\pi(\kappa^+)^k$ and $x_0,\dots,x_{n-1}\in\pi(\kappa^+)$ we have $$(\pi(\kappa^+),B,\in)\models\phi[x_0,\dots,x_{n-1}]$$ there is such a $B$ in $N$. It follows that if $M$ is $\Pi_1^1$-correct about $\kappa^+$, $|M\cap\kappa^+|>|M\cap\kappa|$: $|M\cap\kappa^+|=|\pi(\kappa^+)|$ and $|M\cap\kappa|=|\pi(\kappa)|$, so assume $|\pi(\kappa)|=|\pi(\kappa^+)|$. We can find $B\subseteq\pi(\kappa^+)^2$ that is a cofinal function from $\pi(\kappa)$ to $\pi(\kappa^+)$. Assuming $B\in N$, by elementarity, $\pi^{-1}(B)$ is a cofinal function from $\kappa$ to $\kappa^+$, a contradiction.