Does $\diamondsuit(\kappa)$ provably hold at Woodins or inaccessible Jónssons $\kappa$?

Question

Usually the question whether the diamond principle $\diamondsuit(\kappa)$ holds for some large cardinal $\kappa$ only concerns large cardinal notions of very low consistency (among the weakly compacts). Partly since it does hold for all subtle cardinals, which are only barely stronger than the weakly compacts, and pretty much every large cardinal notion below a weakly compact has been shown to consistently not satisfy it (see Failure of diamond at large cardinals and Ben Neria ('17)).

That subtle cardinals satisfy diamond of course means that almost all large cardinals do satisfy it as well, but there are some strange ones lying around though, including Woodin cardinals and inaccessible Jónsson cardinals. Is anything known about diamond holding for any of these two?

Answer 1

This is a partial answer. I will show that if $\delta$ is Woodin then $\diamondsuit_\delta$ holds.

Claim: Any Woodin cardinal is subtle.

Proof: Let $\delta$ be a Woodin cardinal. Let $\vec{A} = \langle A_\alpha \mid \alpha < \delta\rangle$ be a sequence a sets, $A_\alpha \subseteq \alpha$ and let $C$ be a club in $\delta$. We want to find $\alpha < \beta$ in $C$ such that $A_\alpha = A_\beta \cap \alpha$.

Since $\delta$ is Woodin, there is a cardinal $\kappa < \delta$ which is $\vec{A} \times C$-strong up to $\delta$. Thus, $\kappa \in C$ and there is an elementary emebedding $j\colon V\to M$, such that :

• $\mathrm{crit}\ j = \kappa$,
• $j(\vec{A}) \restriction \kappa + 1 = \vec{A} \restriction \kappa + 1$,

In $M$, $j(\vec{A})(j(\kappa)) \cap \kappa = j(\vec{A})(\kappa) = A_\kappa$ and $\kappa, j(\kappa) \in j(C)$. By elementarity, there is $\alpha < \kappa$ in $C$ such that $A_\alpha = A_\kappa \cap \alpha$.

Answer 2

Assuming that by "inaccessible Jónsson" you meant a regular limit cardinal of uncountable cofinality which is Jónsson; then using the arguments of [1] (Theorem 15 p.115), we have

if $\mathbb{P}$ is c.c.c. and $\kappa$ is Jónsson then for any $V$-generic $G\subset \mathbb{P}$, $V[G]\vDash$ "$\kappa$ is Jónsson".

In particular, if $\kappa$ is Jónsson, and $G\subset \mathbb{P}=\mathsf{Fn}(\kappa^{+}, 2)$ is $V$-generic, then

$V[G] \vDash $ "$\kappa < 2^{\aleph_0}$ and $\kappa$ is Jónsson."

hence $V[G] \vDash \neg \diamondsuit_\kappa$ and $\kappa$ is Jónsson. Moreover, If we started with $\kappa$ which was a regular limit cardinal then the same holds for $\kappa$ in $V[G]$.

[1] Devlin, Keith J., Some weak versions of large cardinal axioms, Ann. Math. Logic 5, 291-325 (1973). ZBL0279.02051.