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

William Adkisson, Omer Ben Neria

TL;DR

The paper analyzes the strength of well-founded ultrafilters on ordinals above a rank Berkeley cardinal and develops a choiceless Prikry forcing framework to control which cardinals can be collapsed or singularized. A central result gives a uniform bound: for any $\gamma$ above $\lambda$ and any well-founded ultrafilter $U$ on a well-ordered domain, there is a surjection from $\bar{\lambda} \times {}^{\gamma}\gamma$ onto $\gamma^X/U$, with $\bar{\lambda}<\lambda$, implying $|\gamma^X/U|<\rho$ for all strong limits $\rho>\lambda$. The work extends Prikry-type forcing to tensor contexts, develops a choiceless Mathias criterion, and shows how tensor Prikry systems can singularize multiple regular cardinals while preserving ZF in symmetric extensions. These methods yield models where all uncountable cardinals are singular and connect choiceless large-cardinal phenomena to precise bounds on forcing behavior, contributing new tools for choiceless set theory and the study of the continuum in such worlds.

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.

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

TL;DR

The paper analyzes the strength of well-founded ultrafilters on ordinals above a rank Berkeley cardinal and develops a choiceless Prikry forcing framework to control which cardinals can be collapsed or singularized. A central result gives a uniform bound: for any above and any well-founded ultrafilter on a well-ordered domain, there is a surjection from onto , with , implying for all strong limits . The work extends Prikry-type forcing to tensor contexts, develops a choiceless Mathias criterion, and shows how tensor Prikry systems can singularize multiple regular cardinals while preserving ZF in symmetric extensions. These methods yield models where all uncountable cardinals are singular and connect choiceless large-cardinal phenomena to precise bounds on forcing behavior, contributing new tools for choiceless set theory and the study of the continuum in such worlds.

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.

Paper Structure

This paper contains 13 sections, 41 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Choiceless Ultrafilters on Ordinals
  4. Tree Prikry Forcing $\mathbb{P}(U)$ in the choiceless setting
  5. Decomposibility and singularizing properties of $\mathbb{P}(U)$
  6. A choiceless version of the Mathias criterion
  7. More about Tensors
  8. When $\mathbb{P}(U_1 \ltimes U_2)$ is equivalent to $\mathbb{P}(U_1) \times \mathbb{P}(U_2)$
  9. Tensor Prikry Systems
  10. Further Applications and Discussions
  11. The first measurable cardinal can be the first strongly inaccessible
  12. The Continuum Problem in models $\mathcal{N}^*$
  13. Strong measures

Key Result

Theorem 1

Assuming there is a rank Berkeley cardinal $\lambda$ there is some $\kappa \geq \lambda$ (which is almost supercompact) such that for every regular cardinal $\delta \geq \kappa$ there is a $\kappa$-complete uniform ultrafilter $U$ on $\delta$.

Theorems & Definitions (89)

  • Theorem 1: Goldberg goldberg:choicelesscardinals
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Definition 9
  • Remark 10
  • ...and 79 more