Is it possible to turn an inaccessible cardinal in $V$ to a successor of a singular cardinal in some forcing extension?



Collapse an inaccessible cardinal to a successor of a singular cardinal
There are several ways to do it:


One is suggested by Noah in his comment.


Another one is to use the supercompact extender based Prikry forcing of Merimovich. See [Supercompact extender based Prikry forcing](http://www2.mta.ac.il/~carmi/Publications/Merimovich%202011%20Supercompact%20extender%20based%20Prikry%20forcing.pdf).


There is also another idea due to Magidor which does the job. See [On a theorem of Magidor](http://math.ipm.ac.ir/~golshani/Papers/Magidor's%20idea.pdf).


**Remark 1.** Note that in all of the above situations, the cardinal which is becoming singular is assumed to be supercompact in the ground model.


**Remark 2.** Using the above ideas and by finding suitable inner models of the final extension, in each case, one can find a pair $(V\_1, V\_2)$
of generic extensions of the ground model in which $\kappa$ is singular in $V\_1$, $\lambda$ in inaccessible in $V\_1$ and in $V\_2$, $\lambda=\kappa^+$.


**Remark 3.** Let me show that some large cardinals are needed to get the result. Assume $V \subset W$ are such that $V \models'' \lambda > \kappa$ is inaccessible'' and $W \models''\kappa$ is singular and $\lambda=\kappa^+$''.


Assume there is no core model for a measurable cardinal and let $K$ be the Dodd-Jensen core model. Pick some $V$-regular cardinal $\mu \in (\kappa, \lambda)$. Then $W \models``cf(\mu)=\theta < \kappa$''. Let $A \subset \mu, otp^W(A)=\theta$ and $sup(A)=\mu.$ By the covering lemma, there exists $B \in K$ with $A \subset B \subset \mu$ and $|B|^W = max\{ \aleph\_1, \theta \}$. But note that then $|B|^K < \kappa,$ so $K \models`` cf(\mu) < \mu$, a contradiction.


