The subcompleteness of diagonal Prikry forcing

TL;DR

This paper proves that certain types of generalized Prikry forcing, which add sequences to measurable cardinals, are subcomplete, extending the understanding of their structural properties in set theory.

Contribution

It establishes the subcompleteness of generalized Prikry forcing and a simplified version called diagonal Prikry forcing, including above specific regular cardinals.

Findings

Generalized Prikry forcing is subcomplete.

Diagonal Prikry forcing is subcomplete.

Subcompleteness holds above certain regular cardinals.

Abstract

Let $D$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $D$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $D$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $D$ is subcomplete above $\mu$, where $\mu$ is any regular cardinal below the first limit point of $D$.