Pseudo-Prikry sequences

TL;DR

This paper extends the theory of pseudo-Prikry sequences, which mimic Prikry-type generic objects, analyzing their existence in outer models and introducing new proof techniques based on PCF theory.

Contribution

It generalizes previous results on pseudo-Prikry sequences and introduces novel methods using PCF theory to analyze their existence in outer models.

Findings

Existence of pseudo-Prikry sequences in outer models.

Introduction of diagonal pseudo-Prikry sequences.

New proof techniques based on PCF theory.

Abstract

We generalize results of Gitik, Dzamonja-Shelah, and Magidor-Sinapova on the existence of pseudo-Prikry sequences, which are sequences that approximate the behavior of the generic objects introduced by Prikry-type forcings, in outer models of set theory. Such sequences play an important role in the study of singular cardinal combinatorics by placing restrictions on the type of behavior that can consistently be obtained in outer models. In addition, we provide results about the existence of diagonal pseudo-Prikry sequences, which approximate the behavior of the generic objects introduced by diagonal Prikry-type forcings. Our proof techniques are substantially different from those of previous results and rely on an analysis of PCF-theoretic objects in the outer model.