Perfect trees and elementary embeddings

Abstract

An important technique in large cardinal set theory is that of extending an elementary embedding j:M→ N between inner models to an elementary embedding j^*:M[G]→ N[G^*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G^* is simply generated by the image of G. But in difficult cases, such as in Woodin’s proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G^* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G^*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin’s theorem as well as the new result that global domination at inaccessibles (the statement that d(κ) is less than 2^κ for inaccessible κ, where d(κ) is the dominating number at κ) is internally consistent, given the existence of 0^#.