Sigma-Prikry forcing II: Iteration Scheme.

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. [ABSTRACT FROM AUTHOR]