Making strong cardinals indestructible by $\leq\kappa$-weakly closed Prikry forcing

This was a talk at the CUNY Set Theory Seminar sometime in 2009.

We did a series of talks joint with Thomas Johnstone on the indestructibility properties of strong cardinals. Thomas presented a folklore result that every strong cardinal can be made indestructible by all $\leq\kappa$-closed forcing. I presented a result of Gitik and Shelah that a strong cardinal can be made indestructible by all $\leq\kappa$-weakly closed forcing with the Prikry condition. A set with two partial orders on it $\langle \mathbb P, \leq,\leq^*\rangle$ satisfies the Prikry condition if for every $p\in \mathbb P$ and every statement $\varphi$ of the forcing language, there exists $q\leq^* p$ such that $q\Vdash\varphi$ or $q\Vdash\neg\varphi$. Let $\alpha$ be a cardinal. We say that $\mathbb P$ is $\leq\alpha$-weakly closed if $\leq^*$ is $\leq\alpha$-closed.

Here are the notes: Making strong cardinals indestructible by $\leq\kappa$-closed forcing (Johnstone)
Making strong cardinals indestructible by $\leq\kappa$-weakly closed Prikry forcing (Gitman)