It is known that if $\delta$ is a Woodin cardinal and $\kappa < \delta$, then the stationary tower forcing $\mathbb Q^\kappa_{<\delta}$ preserves cardinals up to $\kappa$ and forces $\delta = \kappa^+$. Thus if there is a Woodin cardinal $\delta$ then there is a forcing preserving cardinals up to $\aleph_\omega$ and making $\delta = \aleph_{\omega+1}$. But it is also known that $\mathbb Q^\kappa_{<\delta}$ is not $\delta$-c.c.

**Question:** Is there some large cardinal assumption that implies the existence of a cardinal $\kappa > \aleph_{\omega+1}$ and a $\kappa$-c.c. forcing $\mathbb P$ which preserves $\aleph_n$ for finite $n$ and makes $\kappa = \aleph_{\omega+1}$?

I'm no expert on these things, but naively I would suggest two possible approaches: (a) Find a large cardinal $\delta$ that implies the existence of a $\delta$-saturated tower of ideals with similar effects as the stationary tower. (b) Find an inaccessible cardinal $\delta$ with a precipitous tower of ideals of height $\delta$ that preserves the $\aleph_n$'s but actually collapses $\delta$, so that $\delta^+$ is the witness. **Update:** (b) is ruled out by Mohammad's result here.

**Note:** It is consistent relative to large cardinals that there is some $\kappa$-c.c. forcing collapsing a regular $\kappa$ to be $\aleph_{\omega+1}$ while preserving cardinals below $\aleph_\omega$. Namely an $\aleph_{\omega+2}$-saturated ideal on $\aleph_{\omega+1}$, which can be forced from a huge cardinal. But I want to see if it is outright implied by large cardinals, because then it is much easier to combine with other things.

**New Idea:** Foreman-Magidor-Shelah show in "Martin's Maximum Part I" that if $\mu$ is regular and $\kappa > \mu$ is supercompact, then $\mathrm{Col}(\mu,<\kappa)$ forces that $NS_\mu$ is precipitous. I believe this was improved by Goldring to a Woodin cardinal. So perhaps for large $\kappa$, $\mathrm{Col}(\aleph_{\omega+1},<\kappa) * \dot{\mathcal{P}(\aleph_{\omega+1}) / NS}$ does the trick. If we force below $cof(\omega_n)$ for $n > 0$, then we are sure to collapse $\kappa = \aleph_{\omega+2}$ (by a theorem of Shelah), and the whole forcing is $\kappa$-dense, so $\kappa^+$ should be the witness. But the problem is, what happens below $\aleph_\omega$? Despite being precipitous, could forcing with $NS_{\aleph_{\omega+1}}$ actually make $\kappa$ countable? The proof of precipitousness given in the paper is a bit abstract so I have no idea how the generic ultrapower compares to the generic extension.