Superdestructibility: A Dual to Laver's Indestructibility

Abstract. 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,...

• Type: Journal
• Year: 1998
• Where: JSYML
• Authors: Joel David Hamkins, Saharon Shelah
• Keywords: JSYML 1998, Small Forcing, Strong Compactness, κ-closed Forcing