Sigma-Prikry forcing II: Iteration Scheme

TL;DR

This paper develops an iteration scheme for Sigma-Prikry forcing notions, enabling the construction of models where the singular cardinal hypothesis fails and stationary set reflection properties are controlled.

Contribution

It introduces a general iteration scheme for Sigma-Prikry forcing, extending their applicability to complex cardinal and stationary set configurations.

Findings

Constructed models where the singular cardinal hypothesis fails.

Achieved simultaneous reflection of finite families of stationary sets.

Demonstrated the consistency of these properties using Sigma-Prikry iterations.

Abstract

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