Reference request: indestructibility of weakly compacts

Question

I've seen a lot of references to the fact that we can force a weakly compact to be indestructible by Cohen forcing, but after searching for a while haven't had any luck finding a proof of this. Any help with (a) what the proof looks like and (b) if this result is simply folklore, or if there is someone I should attribute this to, would be much appreciated!

Answer 1

Actually I think there is no known way to do this except for the Laver preparation to make a supercompact indestructible (assuming that by "indestructible" you mean indestructible via $\kappa$-directed closed forcings). This comes from Laver's aptly titled "Equiconsistencies at subcompact cardinals."

Answer 2

For indestructibility by adding Cohen subsets to $\kappa$, no extra strength is needed.

If $\kappa$ is weakly compact, then let $\newcommand\P{\mathbb{P}}\P$ be the length $\kappa$ Easton support iteration that uses $\text{Add}(\gamma,1)$ at every inaccessible $\gamma<\kappa$. Let $\newcommand\Q{\mathbb{Q}}\Q=\text{Add}(\kappa,1)$ be the forcing to add a Cohen subset to $\kappa$. In $V[G][g]$, generic for $\P*\Q$, I claim $\kappa$ remains weakly compact by the usual master condition lifting arguments. That is, for any $\kappa$-model $M$ and $j:M\to N$ in $V$, we can lift the embedding to $j:M[G][g]\to M[j(G)][j(g)]$.

The point now is that since adding two Cohen subsets is isomorphic to adding one, it follows that the weak compactness of $\kappa$ is indestructible by adding a Cohen subset over $V[G][g]$.

It follows that the weak compactness of $\kappa$ is indestructible in this model by $\text{Add}(\kappa,\theta)$ for any $\theta$. The reason is that any subset of $\kappa$ added by this forcing is added by a size $\kappa$ piece of it, using only $\kappa$ many of the coordiantes, and so one can thereby reduce to the case $\text{Add}(\kappa,1)$. That is, the embedding witness for any particular subset of $\kappa$ is already present in that extension.

If you want $\kappa$-c.c. indestructibility forcing, then don't force at every $\gamma<\kappa$, but do a lottery-preparation style forcing, and then you get that $\kappa$ is weakly compact in $V[G]$ and indestructible by Cohen forcing there.