Generalized iteration of forcing

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 $\mathcal {P}$.