Thomas Johnstone earned his Ph.D. under my supervision in June, 2007 at the CUNY Graduate Center. Tom likes to get thoroughly to the bottom of a problem, and this indeed is what he did in his dissertation work on the forcing-theoretic aspects of unfoldable cardinals. He seemed to want always to dig deeper, seeking out the unstated general phenomenon behind the results. His characteristic style of giving a seminar talk—pure mathematical pleasure to attend—is to explain not only why the mathematical fact is true, but also why the proof must be the way that it is. Thomas holds a tenure-track position at the New York City College of Technology of CUNY.

Thomas A. Johnstone, “Strongly unfoldable cardinals made indestructible,” Ph.D. dissertation, The Graduate Center of the City University of New York, June 2007.

**Abstract.** I provide indestructibility results for weakly compact, indescribable and strongly unfoldable cardinals. In order to make these large cardinals indestructible, I assume the existence of a strongly unfoldable cardinal $\kappa$, which is a hypothesis consistent with $V=L$. The main result shows that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all ${<}\kappa$-closed forcing which does not collapse $\kappa^{+}$. As strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, I obtain indestructibility for these cardinals also, thereby reducing the large cardinal hypothesis of previously known indestructibility results for these cardinals significantly. Finally, I use the developed methods to show the consistency of a weakening of the Proper Forcing Axiom $\rm PFA$ relative to the existence of a strongly unfoldable cardinal.