Unfoldable cardinals and the GCH

Abstract

Introducing unfoldable cardinals last year, Andres Villaveces ingeniously extended the notion of weak compactness to a larger context, thereby producing a large cardinal notion, unfoldability, with some of the feel and flavor of weak compactness but with a greater consistency strength. Specifically, kappa is theta-unfoldable when for any transitive structure M of size kappa that contains kappa as an element, there is an elementary embedding j:M-->N with critical point kappa for which j(kappa) is at least theta. Define that kappa is fully unfoldable, then, when it is theta-unfoldable for every theta. In this paper I show that the embeddings associated with these unfoldable cardinals are amenable to some of the same lifting techniques that apply to weakly compact embeddings, augmented with methods from the strong cardinal context. Using these techniques, I show by set-forcing over any model of ZFC that any given unfoldable cardinal kappa can be made indestructible by the forcing to add any number of Cohen subsets to kappa. This result contradicts expectations to the contrary that class forcing would be required.