Generalized iteration of forcing

Authors:
M. Groszek and T. Jech

Journal:
Trans. Amer. Math. Soc. **324** (1991), 1-26

MSC:
Primary 03E40; Secondary 03E35, 03E50

DOI:
https://doi.org/10.1090/S0002-9947-1991-0946221-X

MathSciNet review:
946221

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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 is a perfect tree forcing, there is a decomposition such that is countably closed, has the countable chain condition, and adds a -generic set.

**Theorem**. *The mixed-support generalized iteration of perfect tree forcing decompositions along any well-founded partial order preserves* .

**Theorem**. *If* *is consistent, so is* *is arbitrarily large + whenever* *is a perfect tree forcing and* *is a collection of* *dense subsets of* , *there is a* *-generic filter on* .

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DOI:
https://doi.org/10.1090/S0002-9947-1991-0946221-X

Article copyright:
© Copyright 1991
American Mathematical Society