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Generalized iteration of forcing

By M. Groszek and T. Jecr


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

Year: 1991
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