Sigma-Prikry forcing II: Iteration Scheme

Alejandro Poveda, Assaf Rinot, Dima Sinapova

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2 Scopus citations

Abstract

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 Σ-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Σ-Prikry. We showed that given a Σ-Prikry poset ℙ and a ℙ-name for a non-reflecting stationary set T, there exists a corresponding Σ-Prikry poset that projects to ℙ and kills the stationarity of T. In this paper, we develop a general scheme for iterating Σ-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.

Original languageEnglish
Article number2150019
JournalJournal of Mathematical Logic
Volume22
Issue number3
DOIs
StatePublished - 1 Dec 2022

Bibliographical note

Publisher Copyright:
© 2022 World Scientific Publishing Company.

Keywords

  • Iterated forcing
  • Sigma-Prikry forcing
  • singular cardinals hypothesis
  • stationary reflection

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