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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?

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What does "countably closed regular cardinal" mean? – Noah Schweber Jan 31 '13 at 23:50
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 '13 at 0:03
Where can I find a precise definition of "homogeneous forcing"? Thx. – Carlo Von Schnitzel Feb 12 '13 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 '13 at 2:29

1 Answer 1

up vote 4 down vote accepted

I think that the strongly compact Prikry forcing, for $\kappa^+$ strongly compact cardinal $\kappa$ - that forces $\text{cf }\kappa = \text{cf }(\kappa^+)^V = \omega$ without adding bounded subsets to $\kappa$, is homogeneous, but I couldn't prove it or find a reference.

Instead, I'll show something weaker that still implies that the truth value of statements of the form $\phi(a)$ in $V[G]$, where $a\in V$ doesn't depend on the generic $G$: For every $p,q\in \mathbb{P}$, we will find $p^\prime \leq p,\,q^\prime \leq q$ and an automorphism between $\mathbb{P}\restriction p^\prime = \{r \in \mathbb{P} | r \leq p^\prime\}$ and $\mathbb{P}\restriction q^\prime$.

Let $\mathbb{P}$ be the strongly compact Prikry forcing for changing both $\text{cf }\kappa$ and $\text{cf }\kappa^+$ to $\omega$, without adding bounded subsets to $\kappa$.

Recall that a condition in $\mathbb{P}$ is a tree of finite increasing sequences in $P_\kappa (\kappa^+ )$, with finite trunk and above it every element has $U$-many successors ($U$ is fine $\kappa$-complete ultrafilter over $P_\kappa (\kappa^+)$). For the exact definitions and basic properties see section 1.4 in Gitik's chapter in the Handbook.

Let $t$ be the trunk of $p$ and $s$ the trunk of $q$. By narrowing the trees of $p$ and $q$, if necessary, we may assume that for every $r\in p$ above $t$, $(r\setminus t)\cup s \in q$. For every $a\leq p$ define $\pi (a) = \{(r\setminus t)\cup s | r\in a\}$ - this is the required automorphism.

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The paper "Consecutive Singular Cardinals and the Continuum Function" by Apter and Cody also contains similar results for supercompact Prikry forcing. – Mohammad Golshani May 6 '14 at 5:55

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