# How to change the successor of a singular with a Woodin?

I'm looking for references on how to change the successor of a singular cardinal from "more or less" minimal assumptions. If possible, then without adding bounded subsets to the singular either.

In some papers (Gitik's Namba variant paper, for example) I saw footnotes or mentioning that with a Woodin cardinal you can do this using stationary tower forcing. But I couldn't find any references to a written result.

To make this more concrete, let me put the following question.

Suppose that $\aleph_\omega$ is a strong limit cardinal in $V$, and there is a Woodin cardinal. Is it possible to collapse $\aleph_{\omega+1}^V$ without adding bounded subsets to $\aleph_\omega^V$? (I don't mind some structural damage above the new $\aleph_{\omega+1}$, though.)

(I'm saying "a Woodin", but really I mean "approximately a Woodin", in the sense that there's no issue requiring some measurable, or even a second Woodin, above it.)

Any references, or even a sketch of the proof, would be great.

• Using stationary tower forcing, the example is in Jech page 679: you can collapse succesor of a singular and give it any cofinality you want. Consistency strength is exactly that of a Woodin cardinal Oct 12 '14 at 23:25

Assume $GCH$ holds and there exists a proper class of completely Jonsson cardinals and let $\gamma<\lambda$ be regular cardinals. Let $a=\{ \alpha<\lambda: cf(\alpha) =\gamma\},$ and suppose that $a$ belongs to a $\mathbb{P}_{\infty}$ generic $G$ with associated $j:V\to V[G],$ where $\mathbb{P}_{\infty}$ denote the stationary tower class forcing, using arbitrary stationary sets $a$ as conditions.

Then in $V[G],$ the cofinality of $\lambda$ is changed to $\gamma.$ cardinals below $\lambda$ are preserved and if $2^\delta$ is less $\lambda,$ than then no new subsets of $\delta$ are added.

so for example we can change the cofinality of $\aleph_{\omega+1}$ to say $\aleph_{10},$ without adding bounded subsets of $\aleph_\omega.$

This is known as generalized Namba forcing. A good reference is Larson's book "The Stationary Tower: Notes on a Course by W. Hugh Woodin". Another nice reference is the notes given by Sy Friedman $\mathbb{P}_{max}$ and the stationary tower.

Note that by core model theory, such a weird eeffect cannot be achieved if $ZFC$ is preserved by adding $V$ as an additional predicate, without using more than a Woodin cardinal and probably this would need a supercompact cardinal.

Suppose that $γ < λ < κ$ are regular cardinals below a Woodin cardinal $κ$. Forcing with $\mathbb{P}_{<\kappa}$ below the condition $a= \{α < λ : cof(α) = γ\}$, we get that $j[∪b] ∈ j(b)$, i.e., that in $M, j[λ]$ is an ordinal below $j(λ)$ of cofinality $j(γ).$ This means that the critical point of $j$ is $λ$, and that $cof(λ) = γ$ in $M$.

Furthermore, if $α$ is such that $2^\alpha<\lambda$, then all subsets of $α$ in $M$ are in $V.$ Since $V_\kappa^M= V_\kappa^{V[G]}$ , these facts hold in $V[G]$ also.

• (Hm, after the cited paragraph, Larson points out that under additional assumptions on $\kappa$ we can do it without completely Jonsson cardinals, and with $\Bbb P_{<\kappa}$. I suppose that he means that this is the case when $\kappa$ is Woodin, right?) Oct 12 '14 at 3:53
• Wait, so are you suggesting that if $\kappa$ is measurable we can replace $\Bbb P_\infty$ by $\Bbb P_{<\kappa}$? Oct 12 '14 at 4:20
• I think we face some problems. For example if $G$ is $\mathbb{P}_{<\kappa}-$generic, then $V_\kappa$ is not in $Ult(V, G),$ so we may face problems. Another note is that if we want well-foundedness of $Ult(V, G)$ we need more, maybe a Woodin cardinal. I will add more to my answer. Oct 12 '14 at 4:30