# Mildly destroying all worldly cardinals below an inaccessible by forcing

We call a cardinal $\theta$ worldly if $V_\theta \models \mathsf{ZFC}$. If $\kappa$ is an inaccessible cardinal, then we have an unbounded set of worldly cardinals below $\kappa$, in fact even a club set (by considering a chain of elementary submodels of $V_\kappa$).

My question is whether it is possible to destroy the worldly cardinals below an inaccessible cardinal by forcing in a mild way in the following sense: given an inaccessible $\kappa$ can there be a forcing that destroys the worldliness of all worldly cardinals below $\kappa$ while preserving the worldliness of $\kappa$ itself?

Of course some further assumptions will be necessary, since such a forcing must singularize $\kappa$, which implies that $\kappa$ is measurable in some inner model, but I am just asking whether the existence of such a forcing is consistent relative to any appropriate large cardinal assumption.

• Possibly relevant: mathoverflow.net/a/130028/1946. – Joel David Hamkins Feb 27 '17 at 23:03
• Thank you, Joel, for pointing to this question. I was aware of it and your answer, but have thought about it again. However, it seems that your method of destroying a worldly cardinal really depends on it being of countable cofinality in the ground model (and therefore also in the generic extension). I actually do not know under which circumstances we can even expect to be able to change the cofinalities of a club of singular cardinals below an inaccessible $\kappa$ to $\omega$ without destroying the worldliness of $\kappa$ or even collapsing it. – Alexander Block Mar 1 '17 at 16:12
• Yes, it seems difficult. I've been pondering your question over the past few days. One idea I had was like this: start with $\kappa$ measurable and add a Prikry sequence $\kappa_n$. Now, work just between each interval $[\kappa_n,\kappa_{n+1})$ to kill worldly cardinals, e.g. by making them all only $\Sigma_n$-worldly in that interval. But at the same time, try to preserve $V_{\kappa_n}\prec_{\Sigma_n}V_\kappa$. That will hide the sequence and perhaps allow you to keep $\kappa$ worldly, while destroying the ones below. But so far, I haven't managed to fit it all together. – Joel David Hamkins Mar 1 '17 at 16:15
• I've just had another idea for a new strategy, however, which seems to be working. I'll post tonight---I've got to teach right now. – Joel David Hamkins Mar 1 '17 at 16:56
• I would be glad to work on this problem with you during my visit to New York! – Alexander Block Mar 2 '17 at 14:04