TITLE: Collapsing the cardinals between two singular cardinals
QUESTION [6 upvotes]: 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 $\lambda^+$?

REPLY [5 votes]: 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.