Sigma-Prikry forcing I: The Axioms

TL;DR

This paper introduces the class of $ ext{Sigma}$-Prikry forcing notions, demonstrating their applicability to singular cardinals and stationary set manipulation, and develops an iteration scheme to achieve specific set-theoretic properties.

Contribution

It defines $ ext{Sigma}$-Prikry forcing, shows their relevance to known Prikry-type notions, and develops an iteration method to control stationary sets and continuum size.

Findings

$ ext{Sigma}$-Prikry notions encompass many known Prikry-type forcings.

Existence of $ ext{Sigma}$-Prikry extensions that kill stationarity of certain sets.

A forcing extension with a strong limit cardinal, simultaneous reflection, and continuum size $\kappa^{++}$.

Abstract

We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma$-Prikry. We show that given a $\Sigma$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $\Sigma$-Prikry poset that projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $\Sigma$-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $\kappa$ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a cofinality-preserving forcing extension in which $\kappa$ remains a strong limit, every finite collection of stationary subsets of $\kappa^+$ reflects simultaneously, and $2^\kappa=\kappa^{++}$.