Sigma-Prikry forcing II: Iteration Scheme

Joint work with Alejandro Poveda and Dima Sinapova.

Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\mathrm{\Sigma }$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\mathrm{\Sigma }$-Prikry. We proved that given a $\mathrm{\Sigma }$-Prikry poset $\mathbb{P}$ and a $\mathbb{P}$-name for a non-reflecting stationary set $T$, there exists a corresponding $\mathrm{\Sigma }$-Prikry poset that projects to $\mathbb{P}$ and kills the stationarity of $T$.

In this paper, we develop a general scheme for iterating $\mathrm{\Sigma }$-Prikry posets, as well as verify that the Extender Based Prikry Forcing is $\mathrm{\Sigma }$-Prikry.

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, yielding a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

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