On the strength of ultrafilters above choiceless large cardinals and their Prikry forcings

TL;DR

This paper investigates the strength of ultrafilters above choiceless large cardinals, extends Prikry forcing theory, and constructs models where all uncountable cardinals are singular, advancing understanding in choiceless set theory.

Contribution

It provides bounds on large cardinal strength from ultrafilters, extends Prikry forcing, and introduces tensor Prikry systems for new consistency results.

Findings

Ultrafilters above choiceless large cardinals have bounded large cardinal strength.

Extended Prikry forcing limits the collapse and singularization of cardinals.

Constructed a model where all uncountable cardinals are singular.

Abstract

We study the strength of well-founded ultrafilters on ordinals above choiceless large cardinals and their associated Prikry forcings. Gabriel Goldberg showed that all but boundedly many regular cardinals above a rank Berkeley cardinal carry well-founded uniform ultrafilters. We prove several bounds on the large cardinal strength that is witnessed by such ultrafilters. We then extend the theory of Prikry forcing in this context and place limits on the cardinals that can be collapsed or singularized. Finally, we develop the notion of a tensor Prikry system, and use it to give new constructions for several consistency results in choiceless set theory. In particular, we build a new model in which all uncountable cardinals are singular.