singularize the least inaccessible?

Question

Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while preserving all cardinals?

Answer 1

In the paper "On Lowenheim-Skolem-Tarski numbers for extension of first order logic", by Magidor and Vaananen, in Theorem 21 they state that it is consistent, relative to the existence of a supercompact cardinal, that the Lowenheim-Skolem-Tarski number of $L(I)$ is the first inaccessible cardinal, were $L(I)$ is the extension of the first order logic with a quantifier of "equicardinality".

An important ingredient in the proof is the following observation (which I state in a slightly different way from the original paper):

Let $\kappa$ be a measurable cardinal. Let $\mathbb{NM}$ be the forcing that adds a club $D$ through the singular cardinals below $\kappa$ using bounded approximations. I will assume that the non limit points of the club are inaccessible cardinals. Let $Col$ be the forcing that collapses all cardinal between the any successor of a point in $D$ and the next point of the $D$, with Easton support. So $\mathbb{NM}\ast Col$ will force $\kappa$ to be the first inaccessible.

On the other hand, in $V$ let $\mathbb{P}$ be a Prikry type forcing that adds a cofinal $\omega$-sequence at $\kappa$, $\{\eta_0, \eta_1, \dots\}$ and pick a sequence of conditions in $\mathbb{NM}$, $\{p_0, p_1,\dots\}$ such that $p_i \subseteq [\gamma_i, \gamma_{i+1})$. So $\mathbb{P}$ singularizes $\kappa$ and adds a club $\tilde{D} = \bigcup p_n$ through the singular cardinals below $\kappa$. Let $\tilde{Col}$ be the forcing that collapses any cardinal between point in $\tilde{D}$ as above, again with Easton support. Note that this time, as $\kappa$ is singular when we define $\tilde{Col}$, the support is unbounded in $\kappa$.

Lemma: $\mathbb{NM}\ast Col$ forces $\kappa$ to be the first inaccessible cardinal. $\mathbb{P}\ast \tilde{Col}$ forces $\text{cf } \kappa = \omega$, and doesn't collapse cardinals above $\kappa$.

Theorem: $\mathbb{P}\ast \tilde{Col}$ adds a generic filter for $\mathbb{NM}\ast Col$, and the quotient forcing, $\mathbb{R}$, changes the cofinaly of $\kappa$ to $\omega$ and doesn't add bounded subsets of $\kappa$.

A variant of this theoerm is proved in the paper of Magidor an Vaananen, during the proof of Theorem 21. The main difference is that they assume there that $\kappa$ is $\kappa^+$-strongly compact in order to obtain a cleaner definition on the generic for $\mathbb{NM}$ that is obtained from the Prikry forcing. Magidor showed me that the same can be achieved using only a measurable, and this is the version that I sketched above.