Collapse an inaccessible cardinal to a successor of a singular cardinal

Question

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

Answer 1

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.

There is also another idea due to Magidor which does the job. See On a theorem of Magidor.

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.