Indestructibility and destructible measurable cardinals

Say that κ’s measurability is destructible if there exists a <κ-closed forcing adding a new subset of κ which destroys κ’s measurability. For any δ, let λδ =df The least beth fixed point above δ. Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, δ is not δ+ strongly compact, and δ’s measurability is destructible when forcing with partial orderings having rank below λδ} is unbounded in κ. On the other hand, under the same hypotheses, A2 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, δ is not δ+ strongly compact, and δ’s measurability is indestructible when forcing with either Add(δ, 1) or Add(δ, δ+)} is unbounded in κ as well. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two distinct models in which either A1 = ∅ or A2 = ∅. In each of these models, both of which have restricted large cardinal structures above κ, every measurable cardinal δ which is not a limit of measurable cardinals is δ+ strongly compact, and there is an indestructibly supercompact cardinal κ. In the model in which A1 = ∅, every measurable cardinal δ which is not a limit of measurable cardinals is ∗2010 Mathematics Subject Classifications: 03E35, 03E55. †