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