Indestructibility and destructible measurable cardinals

Abstract

Say that ${\kappa}$’s measurability is destructible if there exists a < ${\kappa}$-closed forcing adding a new subset of ${\kappa}$ which destroys ${\kappa}$’s measurability. For any δ, let λ _δ =_df The least beth fixed point above δ. Suppose that ${\kappa}$ is indestructibly supercompact and there is a measurable cardinal λ > ${\kappa}$. It then follows that ${A_{1} = \{\delta < \kappa \mid \delta}$ 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 λ _δ } is unbounded in ${\kappa}$. On the other hand, under the same hypotheses, ${A_{2} = \{\delta < \kappa \mid \delta}$ 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 ${\kappa}$ as well. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two distinct models in which either ${A_{1} = \emptyset}$ or ${A_{2} = \emptyset}$. In each of these models, both of which have restricted large cardinal structures above ${\kappa}$, every measurable cardinal δ which is not a limit of measurable cardinals is δ ⁺ strongly compact, and there is an indestructibly supercompact cardinal ${\kappa}$. In the model in which ${A_{1} = \emptyset}$, every measurable cardinal δ which is not a limit of measurable cardinals is <λ _δ strongly compact and has its <λ _δ strong compactness (and hence also its measurability) indestructible when forcing with δ-directed closed partial orderings having rank below λ _δ . The choice of the least beth fixed point above δ is arbitrary, and other values of λ _δ are also possible.