Sigma-Prikry forcing II: Iteration Scheme

Journal of Mathematical Logic 22 (3) (2022)
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Abstract

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [math]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [math]-Prikry. We showed that given a [math]-Prikry poset [math] and a [math]-name for a non-reflecting stationary set [math], there exists a corresponding [math]-Prikry poset that projects to [math] and kills the stationarity of [math]. In this paper, we develop a general scheme for iterating [math]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

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Extender based forcings.Moti Gitik & Menachem Magidor - 1994 - Journal of Symbolic Logic 59 (2):445-460.
Blowing up the power of a singular cardinal.Moti Gitik - 1996 - Annals of Pure and Applied Logic 80 (1):17-33.

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