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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.

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  • $\begingroup$ Mmmm... chain conditions. We [Yair and I] could have totally tried to tackle that in our work. Why didn't you ask that two months ago? :-) $\endgroup$ – Asaf Karagila Feb 11 '15 at 11:00
  • $\begingroup$ Sorry, just thought of it. :-) $\endgroup$ – Monroe Eskew Feb 11 '15 at 11:01
  • $\begingroup$ Although maybe it's possible to squeeze out of what we did something related to chain conditions. Let me think about it until Yair gets here. :-P $\endgroup$ – Asaf Karagila Feb 11 '15 at 11:03
  • $\begingroup$ By the way, what happens if you take a Woodin which is a limit of Woodin cardinals, and iterate stationary towers up to that limit, with bounded support or something like that. $\endgroup$ – Asaf Karagila Feb 11 '15 at 11:23
  • $\begingroup$ Good question, I don't know. Does the preservation of $\aleph_n$'s go through limit stages? $\endgroup$ – Monroe Eskew Feb 11 '15 at 11:26
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This may not be an answer but it is longer than be a comment:

Starting from a supercompact cardinal $\kappa$ an inaccessible $\lambda$ above it, we can construct a model $M$ of ZFC in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.$ The model in an intermediate submodel of a supercompact Prikry forcing with suitable collapses.

I think using an analysis similar to Foreman-Woodin's paper "GCH can fail everywhere", we can show that this intermdiate submodel $M$ is a $\lambda-c.c.$ extension of the ground model.

Now find another intermediate submodel $N \subset M$ of the ground model which is essentially the Prikry extension of the ground which makes $\kappa$ into $\aleph_\omega,$ preserves cardinals above $\kappa$ and it makes the cardinal structure below $\aleph_\omega$ the same as in $M$.

Now consider $M$ as a generic extension of $N$, which makes $\lambda$ into $\aleph_{\omega+1}$ and is $\lambda-c.c.$ extension of $N$.

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  • $\begingroup$ When I asked if it's even consistent, this was the sort of idea that I had in mind. Thanks for reinforcing my intuition! $\endgroup$ – Asaf Karagila Feb 11 '15 at 17:09
  • $\begingroup$ @AsafKaragila Your welcome, in fact I learnt such an idea from Prof. Magidor; your supervisor!. $\endgroup$ – Mohammad Golshani Feb 11 '15 at 18:52
  • $\begingroup$ How do you show the model $N$ exists? $\endgroup$ – Monroe Eskew Feb 12 '15 at 6:14
  • $\begingroup$ By defining over the ground model a Prirky forcing with suitable collapses, and then defining a weak projection from supercompact prikry forcing into it (to guarantee that the $\omega-$sequence added by them and the collapses are the same, so that the cardinal structure below $\aleph_\omega$ is the same in both $M$ and $N$). $\endgroup$ – Mohammad Golshani Feb 12 '15 at 6:18
  • $\begingroup$ @MohammadGolshani: You results is stronger: Let $N = V[G]$, where $G$ is a generic for the standard Prikry forcing on $\kappa$ with collapses (the normal measure on $\kappa$ projected from the measure on $P_\kappa \lambda$). In this model, for every $\mu$, let $\mathbb{R}$ be the quotient between the supercompact Prikry forcing with collapses using the measure on $P_\kappa \mu^+$ (from the ground model), and $G$. This forcing is $\mu$-centered if $\mu > \kappa$. $\endgroup$ – Yair Hayut Feb 12 '15 at 7:41

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