Laver indestructibility and the class of compact cardinals

Journal of Symbolic Logic 63 (1):149-157 (1998)
Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every κ ∈ K is a supercompact cardinal indestructible under κ-directed closed forcing, and every κ a measurable limit point of K is a strongly compact cardinal indestructible under κ-directed closed forcing not changing ℘(κ). We then derive as a corollary a model for the existence of a strongly compact cardinal κ which is not κ + supercompact but which is indestructible under κ-directed closed forcing not changing ℘(κ) and remains non-κ + supercompact after such a forcing has been done
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DOI 10.2307/2586593
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Joel David Hamkins (2000). The Lottery Preparation. Annals of Pure and Applied Logic 101 (2-3):103-146.

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