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

2025/11/10 by William Adkisson, Adkisson, William, Omer Ben Neria +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2511.07622

openalex publication_date 2025/11/10 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28

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.

Citations

• The first measurable can be the first inaccessible cardinal
• Measurable cardinals and choiceless axioms
• On products of ultrafilters
• Extenders under ZF and constructibility of rank-to-rank embeddings
• Choiceless Löwenheim-Skolem property and uniform definability of grounds

Related

• A Small Ultrafilter Number at Every Singular Cardinal
• Some combinatorial properties of Ultimate L and V
• The combinatorics of reasonable ultrafilters
• Applications of the Magidor Iteration to Ultrafilter Theory
• A small ultrafilter number at smaller cardinals