Prikry-type forcing and the set of possible cofinalities

TL;DR

This paper investigates how Prikry-type forcing and the presence of measurable cardinals affect the properties of the set of possible cofinalities of certain intervals of regular cardinals, revealing cases where these properties fail.

Contribution

It demonstrates that the set of possible cofinalities can lack good properties under Prikry-type forcing and measurable cardinals, contrasting with known results for progressive intervals.

Findings

The set pcf(A) can lack good properties under Prikry-type forcing.

Measurable cardinals can disrupt the usual properties of pcf(A).

Certain intervals of regular cardinals exhibit no good pcf properties in these contexts.

Abstract

It is known that the set of possible cofinalities $\mathrm{pcf}(A)$ has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that $\mathrm{pcf}(A)$ has no good properties in the presense of a measurable cardinal, or in generic extensions by Prikry-type forcing.