The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Blue, Douglas, Larson, Paul, Sargsyan, Grigor
Institutions: University of Pittsburgh, Miami University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7129001925