Ultraexacting above extendible implies consistency of rank-Berkeley cardinals

Establish that, assuming ZFC, the existence of an ultraexacting cardinal λ together with an extendible cardinal κ < λ suffices to prove the consistency of ZF with a rank-Berkeley cardinal by constructing a set-sized model of ZF containing a rank-Berkeley cardinal.

Background

The paper introduces exacting and ultraexacting cardinals, shows their consistency relative to I0, and analyzes their interaction with extendible cardinals and HOD. A central technical result (Theorem ConZF) establishes that ZF with a C^{(3)}-Reinhardt cardinal and a supercompact cardinal above the supremum of the critical sequence proves the consistency of ZFC with an ultraexacting cardinal that is a limit of extendible cardinals.

Building on these results and the demonstrated refutation of the Weak HOD Conjecture under suitable large-cardinal hypotheses, the authors conjecture a strengthening: that merely having an ultraexacting cardinal above an extendible cardinal within ZFC already yields the consistency of ZF with rank-Berkeley cardinals (large cardinals known to be beyond the Axiom of Choice). This would significantly lower the consistency strength required to reach choiceless large cardinals from ZFC assumptions involving ultraexacting and extendible cardinals.