Is it possible, starting from any large cardinal assumption, to find a *countably closed* forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash\_\mathbb{P} "\kappa = \lambda^+$ and $\lambda$ is singular"?



Inaccessible becomes successor of singular
No. The following theorem is from a work in progress by Yair Hayut and myself.



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> **Theorem.** If $\Bbb P$ is a proper forcing, and it changes the cofinality of $\kappa$ to $\mu>\omega$, then $\Bbb P$ adds a surjection from $\mu$ onto $\kappa$.
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Now suppose that you had such countably closed $\Bbb P$, it is certainly proper. And it changes the cofinality of $(\lambda^+)^V$ to be something which is smaller than $\lambda$, and therefore collapses $(\lambda^+)^V$, and as a consequence it must collapse $\lambda$ as well.


(It should be remarked that a countably closed forcing cannot change the cofinality of something $\omega$ anyway.)


