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Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$?

If the answer to question 1 is affirmative, it will probably involve a forcing construction. It is natural to ask whether a similar situation can occur "naturally":

Question 2: Does some large cardinals assumption imply that there is a pair of singular cardinals $\mu < \lambda$ and a forcing notion collapsing $\lambda$ to $\mu$ without collapsing $\mu$ or $\lambda^+$?

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2 Answers 2

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For question 2; let $\kappa$ be a supercompact cardinal, and consider the diagonal Prikry forcing $\mathbb{P}$ to change the cofinality of $\kappa$ to $\omega$ and collapses all cardinals in $(\kappa^+, \kappa^{+\omega})$ into $\kappa$ and preserving all other cardinals. Let $V[G]$ be the resulting extension. But note that then there is some intermediate submodel $V[H], V \subset V[H] \subset V[G],$ which is essentially the ordinary Prikry extension of $V$ for changing the cofinality of $\kappa$ and preserving all cardinals.

Now $V[G]$ considered as a generic extension of $V[H]$ has the required property.

For question 1 add collapses to make $\kappa=\aleph_\omega,$ and define $V[H]$ similarly so that the cardinal structure of $V[H]$ and $V[G]$ below $\kappa$ is the same.

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It is possible to get a stronger result. Start with a supercompact cardinal and force with supercompact Radin forcing with a suitable measure sequences over $P_\kappa(\kappa^{+\kappa}).$ The forcing preserves $\kappa$ inaccessible, adds a club $C$ of former regulars into $\kappa,$ and for any $\alpha$, a limit point of $C$, it collapses all cardinals in $(\alpha, \alpha^{+\alpha}]$ into $\alpha$ but preserves all other cardinals below $\kappa.$ Call the extension $V[G].$ We can arrange a submodel $V[H], V \subset V[H] \subset V[G],$ such that $V[H]$ is a cardinal preserving extension of $V$, but $C \in V[H].$ Now if $\alpha$ is a singular limit point of $C$ in $V[G]$, then it is the same in $V[H]$, and passing from $V[H]$ to $V[G],$ the forcing collapses the singular cardinal $\alpha^{+\alpha}$ into the singular $\alpha,$ but it preserves $\alpha, \alpha^{+\alpha+1}$.

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  • $\begingroup$ Your first paragraph in not an answer for question 2 since you're singularizing $\kappa$. Also, I think that you can't use guiding generic in the collapsing phase since the guiding generic for the supercompact forcing will not be compatible with the one for the regular Prikry forcing. So the collapsing part will not be $\kappa^{+\omega+1}$-c.c., but it won't collapse it, so it does answer my question. $\endgroup$
    – Yair Hayut
    Jul 28, 2015 at 20:37
  • $\begingroup$ But in $V[H], \kappa$ is singular of cofinality $\omega,$ and passing from $V[H]$ to $V[G],$ we are collapsing all cardinals in the interval $(\kappa, \kappa^{+\omega}]$ without collapsing other cardinals, and both of this cardinals are already singular in $V[H].$ $\endgroup$ Jul 29, 2015 at 2:47
  • $\begingroup$ Thanks. I agree that $V[H]$ solves question 1 (and the new version of $V[H]$ actually suggests that a wild further strengthening is possible, so this solution is much stronger than what I expected). But in $V$ we don't know if there is such pair of cardinals, so the existence of the collapsing forcing does not follow merely from the large cardinals assumptions. $\endgroup$
    – Yair Hayut
    Jul 29, 2015 at 5:55
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For question 2, some large cardinals imply the existence of such a forcing. Suppose $\kappa$ is 2-huge, with $j : V \to M$ an elementary embedding, $\lambda = j(\kappa)$, $\theta = j(\lambda)$, and $M^\theta \subseteq M$. Then using the embedding, we see $S = \{ x \subseteq \lambda^{+\omega+1} : (\forall \alpha \leq \omega+1) \mathrm{ot}(x \cap \lambda^{+\alpha}) = \kappa^{+\alpha} \}$ is stationary. A standard argument gives that the following set is also stationary: $T = \{ x \subseteq \lambda^{+\omega+1} : \mathrm{ot}(x \cap \lambda^{+\omega+1}) = \kappa^{+\omega+1}$ and $\kappa^{+\omega} \subseteq x \}$. Also, there is a Woodin cardinal $\delta > \lambda$. Forcing with the stationary tower up to $\delta$ below $T$ will produce a generic embedding $i : V \to N \subseteq V[G]$ with critical point $\kappa^{+\omega+1}$ and $i(\kappa^{+\omega+1}) = \lambda^{+\omega+1}$, and $N^\delta \subseteq N$. This has the desired collapsing effects.

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  • $\begingroup$ Thanks. Is it possible to reduce the large cardinal requirements? As it seems now, you're using "huge with closure $j(\kappa)^{+\omega+1}$". Can you reduce it to something less than a huge cardinal? $\endgroup$
    – Yair Hayut
    Jul 28, 2015 at 20:54
  • $\begingroup$ Good question. Are there weaker large cardinals that imply the kind of Chang conjectures used here? $\endgroup$ Jul 28, 2015 at 21:37
  • $\begingroup$ Looking a bit in the literature, it looks like you are right and there is no known way to obtain this CC case from less that this kind of large cardinals. $\endgroup$
    – Yair Hayut
    Jul 29, 2015 at 17:03
  • $\begingroup$ Here's an idea for lowering the strength slightly. If $\kappa$ is huge with closure $j(\kappa)^{+\omega+1}$, then the ultrapower $M$ also sees $(j(\kappa)^{+\omega+1},j(\kappa)^{+\omega}) \twoheadrightarrow (\kappa^{+\omega+1},\kappa^{+\omega}) $. So this reflects to many pairs below $\kappa$. $\endgroup$ Jul 29, 2015 at 19:24
  • $\begingroup$ By the way, you can also force a similar situation around $\aleph_\omega$, giving another approach to question 1. See Foreman's handbook chapter, section 7. But Mohammad's approach uses weaker assumptions. $\endgroup$ Jul 29, 2015 at 22:40

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