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JSYML
1998

Superdestructibility: A Dual to Laver's Indestructibility

9 years 11 months ago
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,...
Joel David Hamkins, Saharon Shelah
Added 22 Dec 2010
Updated 22 Dec 2010
Type Journal
Year 1998
Where JSYML
Authors Joel David Hamkins, Saharon Shelah
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