Generalized iteration of forcing

Abstract

ABSTRACT. Generalized iteration extends the usual notion of iterated forcing from iterating along an ordinal to iterating along any partially ordered set. We consider a class of forcings called perfect tree forcing. The class includes Axiom A forcings with a finite splitting property, such as Cohen, Laver, Mathias, Miller, Prikry-Silver, and Sacks forcings. If go is a perfect tree forcing, there is a decomposition ~ * [Jf such that ~ is countably closed, [Jf has the countable chain condition, and ~ * [Jf adds a go-generic set. Theorem. The mixed-support generalized iteration of perfect tree forcing decompositions along any well-founded partial order preserves CUI • Theorem. If ZFC is consistent, so is ZFC + 2 0J is arbitrarily large + whenever go is a perfect tree forcing and £g is a collection of CUI dense subsets of go, there is a £g-generic filter on go. O.