A Note on Indestructibility and Strong Compactness

摘要

If K < λ are such that K is both supercompact and indestructible under K-directed closed forcing which is also (K~+, ∞)-distributive and λ is 2~supercompact, thenby a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)],{δ< K|δis δ~+stronglycompact yet b is not 6+ supercompact} must be unbounded in We show that the largecardinal hypothesis on A is necessary by constructing a model containing a supercompactcardinal in which no cardinal δ > K is 2~δ = δ~+supercompact,k'ssupercompactness isindestructible under ic-directed closed forcing which is also (pc+, co)-distributive, and forevery measurable cardinal δ, δ is δ~+strongly compact iff 5 is δ~+