Unfoldable Cardinals and The GCH

Unfoldable cardinals are preserved by fast function forcing and the Laver-like preparations that fast functions support. These iterations show, by set-forcing over any model of ZFC, that any given unfoldable cardinal κ can be made indestructible by the forcing to add any number of Cohen subsets to κ. 1 Unfoldable Cardinals In introducing unfoldable cardinals last year, Andres Villaveces [Vil98] 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. In this paper I will 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 some methods from the strong cardinal context. Using these techniques, I will show by set-forcing that any unfoldable cardinal κ can be made indestructible by the forcing to add any number of Cohen subsets to κ. Villaveces defines that a cardinal κ is θ-unfoldable when for every transitive model M of size κ with κ ∈ M there is an elementary embedding j : M → N with critical point κ and j(κ) ≥ θ.1 The cardinal κ is fully unfoldable when it is θ-unfoldable for every ordinal θ. A lower bound on the consistency strength of unfoldability is My research has been supported in part by grants from the PSC-CUNY Research Foundation and the NSF. I would particularly like to thank Philip Welch for suggesting this line of research to me and for his helpful discussions concerning it during my recent return to Japan in July, 1999. And I would like also to thank Kobe University for their generous support during that trip. Specifically, the College of Staten Island of CUNY and the CUNY Graduate Center. Actually, Villaveces defines that κ is θ-unfoldable if and only if it is inaccessible and for every S ⊆ κ there is an Ŝ and a transitive N of height at least θ such that 〈Vκ,∈, S 〉 ≺ 〈N,∈, Ŝ 〉. He then proves this definition equivalent to the embedding characterization, which I prefer to take as the basic notion.