Indestructibility and stationary reflection

If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ-strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a Mahlo cardinal in which the least supercompact cardinal κ is also the least strongly compact cardinal, κ’s strongness is indestructible under κ-strategically closed forcing, κ’s supercompactness is indestructible under κ-directed closed forcing not adding any new subsets of κ, and δ is Mahlo and reflects stationary sets iff δ is weakly compact. In this model, no strong cardinal δ < κ is indestructible under δ-strategically closed forcing. It therefore follows that it is relatively consistent for the least strong cardinal κ whose strongness is indestructible under κ-strategically closed forcing to be the same as the least supercompact cardinal, which also has its supercompactness indestructible under κ-directed closed forcing not adding any new subsets of κ. ∗2000 Mathematics Subject Classifications: 03E35, 03E55. †