The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals

arXiv (Cornell University) 2026 Preprint Open access

Read full text
Authors: Douglas Blue, Paul G. Larson, Grigor Sargsyan
DOI: 10.48550/arxiv.2602.13077
Cited by: 0

Abstract

We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC.

Fields of Study & Subjects

Advanced Topology and Set Theory Mathematical and Theoretical Analysis Computability Logic AI Algorithms

Publication details

Journal / source
arXiv (Cornell University)
Publication date
13 February 2026
Language
EN
Document type
Preprint