Bea Adam-Day (University of Leeds)

Indestructibility and $C^{(n)}$-supercompact cardinals

In the 70's Laver showed that a supercompact cardinal $\kappa$ may be made indestructible by a suitable class of forcings—namely, after a preparatory forcing, the supercompactness of $\kappa$ will not be destroyed by any further $<\kappa$-directed closed forcing. Many indestructibility results have since been written, as well as those demonstrating the impossibility of indestructibility (or even preservation) of many large cardinals. In this talk we will consider the case of $C^{(n)}$-supercompact cardinals—a stronger and more slippery variant of supercompact cardinals—and how they can be made indestructible for $n\leq 2$.