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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 $ \mathcal{P}$ is a perfect tree forcing, there is a decomposition $ \mathcal{Q} * \mathcal{R}$ such that $ \mathcal{Q}$ is countably closed, $ \mathcal{R}$ has the countable chain condition, and $ \mathcal{Q} * \mathcal{R}$ adds a $ \mathcal{P}$-generic set.

Theorem. The mixed-support generalized iteration of perfect tree forcing decompositions along any well-founded partial order preserves $ {\omega _1}$.

Theorem. If $ {\text{ZFC}}$ is consistent, so is $ {\text{ZFC + }}{{\text{2}}^\omega }$ is arbitrarily large + whenever $ \mathcal{P}$ is a perfect tree forcing and $ \mathcal{D}$ is a collection of $ {\omega _1}$ dense subsets of $ \mathcal{P}$, there is a $ \mathcal{D}$-generic filter on $ \mathcal{P}$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-0946221-X
Article copyright: © Copyright 1991 American Mathematical Society

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