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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 $\aleph\_{\omega\cdot 2 + 1}$? 
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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":



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> **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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Collapsing the cardinals between two singular cardinals
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 $\alpha, \alpha^{+\alpha+1}$.


