Superdestructibility: A Dual to Laver's Indestructibility

After small forcing, any <κ-closed forcing will destroy the supercompactness and even the strong compactness of κ. In a delightful argument, Laver [L78] proved that any supercompact cardinal κ can be made indestructible by <κ-directed closed forcing. This indestructibility, however, is evidently not itself indestructible, for it is always ruined by small forcing: in [H96] the first author recently proved that small forcing makes any cardinal superdestructible; that is, any further <κ-closed forcing which adds a subset to κ will destroy the measurability, even the weak compactness, of κ. What is more, this property holds higher up: after small forcing, any further <κ-closed forcing which adds a subset to λ will destroy the λ-supercompactness of κ, provided λ is not too large (his proof needed that λ < אκ+δ, where the small forcing is <δ-distributive). In this paper, we happily remove this limitation on λ, and show that after small forcing, the supercompactness of κ is destroyed by any <κ-closed forcing. Indeed, we will show that even the strong compactness of κ is destroyed. By doing so we answer the questions asked at the conclusion of [H96], and obtain the following attractive complement to Laver indestructibility: Main Theorem. After small forcing, any <κ-closed forcing will destroy the supercompactness and even the strong compactness of κ. We will provide two arguments. The first, similar to but generalizing the Superdestruction Theorem of [H96], will show that supercompactness is destroyed; the second, by a different technique, will show fully that strong compactness is destroyed. Both arguments will rely fundamentally on the Key Lemma, below, which was proved in [H96]. Define that a set or sequence is fresh over V when it is not in V but every initial segment of it is in V . † The first author’s research has been supported in part by the College of Staten Island and a grant from The City University of New York PSC-CUNY Research Award Program. ‡ The second author’s research has been supported by The Israel Science Foundation, administered by the Israel Academy of Sciences and Humanities. This is publication 618 in his independent numbering system.