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Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly inaccessible. Can there be a homogeneous notion of forcing that makes $\text{cof}(\kappa^{+V}) < \kappa$ without adding any bounded subsets of $\kappa$?

If there is a Woodin cardinal above $\kappa$ then the stationary tower forcing could do this except that it is (probably) not homogeneous.

If there is a forcing notion as desired then I believe the results of the paper "Stacking mice" would give a non-domestic mouse, so some large cardinals would be required to show that such a forcing exists. Can we get one from, e.g. a supercompact cardinal?

edited Feb 1 at 0:47

asked Jan 31 at 21:11

Trevor Wilson
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What does "countably closed regular cardinal" mean? – Noah S Jan 31 at 23:50

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Perhaps he means $\delta^\omega\lt\kappa$ whenever $\delta\lt\kappa$. (But he may mean simply that $\kappa^\omega=\kappa$.) – Joel David Hamkins Feb 1 at 0:03

Where can I find a precise definition of "homogeneous forcing"? Thx. – alephomega Feb 12 at 8:49

@alephomega When I use that phrase I usually mean "almost homogeneous forcing", which means that for any two conditions $p$ and $q$ in the forcing poset $\mathbb{P}$ there is an automorphism $\pi$ of $\mathbb{P}$ such that $\pi(p)$ is compatible with $q$. The only consequence of this that I am interested in is that the theory of the forcing extension with parameters from the ground model does not depend on the choice of generic filter. – Trevor Wilson Feb 15 at 2:29

Homogeneous Namba-like forcing

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