A Note on Indestructibility and Strong Compactness

Abstract

Summary. If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ + , ∞)-distributive and λ is 2 λ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], {δ < κ | δ is δ + strongly compact yet δ is not δ + supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is 2 δ = δ + supercompact, κ's supercompactness is indestructible under κ-directed closed forcing which is also (κ + , ∞)-distributive, and for every measurable cardinal δ, δ is δ + strongly compact iff δ is δ + supercompact. Introduction and preliminaries. In [3], it was shown (see Theorem 5) that if κ < λ are such that κ is indestructibly supercompact and λ is 2 λ supercompact, then {δ < κ | δ is δ + strongly compact yet δ is not δ + supercompact} must be unbounded in κ. The only use of indestructibility in this proof is that κ remains supercompact after forcing with the partial ordering which first (if necessary) makes 2 λ = λ + and 2 λ + = λ ++ and then does a reverse Easton iteration of length λ which adds a nonreflecting stationary set of ordinals of cofinality κ to each measurable cardinal in a final segment of the open interval (κ, λ). Thus, we actually have the following result.