Berkeley Cardinals and Vopěnka's Principle
Marwan Salam Mohammd
TL;DR
The paper studies large-cardinal notions in $ ext{ZF}$ without the Axiom of Choice, introducing $n$-choiceless extendible and related cardinals and linking them to Vopěnka's Principle $ ext{$ ext{VP}$}$. It proves that $ ext{$ ext{VP}$}$ restricted to certain definability classes is equivalent to the existence of many $n$-choiceless extendible cardinals, mirroring Bagaria's $C^{(n)}$-cardinals results in a choiceless setting, and establishes several equiconsistency results. Berkeley and rank-Berkeley cardinals are defined and tied to $ ext{$ ext{VP}$}$, yielding equiconsistency between $ ext{ZF}+ ext{$ ext{VP}$}^ ext{ω}+ eg ext{$ ext{VP}$}$ and $ ext{ZF}+ ext{BC}$, with analogous connections at higher definability levels via $ ext{OR}$-Mahlo considerations. Together, these results illuminate the structure and strength of Vopěnka-type principles without the Axiom of Choice and connect them to class-embedding phenomena and forcing-like analyses in the choiceless realm.
Abstract
We introduce "$n$-choiceless" supercompact and extendible cardinals in Zermelo-Fraenkel set theory without the Axiom of Choice. We prove relations between these cardinals and Vopěnka's Principle similar to those of Bagaria's work in his papers "$C^{(n)}$-Cardinals" and "More on the Preservation of Large Cardinals Under Class Forcing." We use these relations to characterize Berkeley cardinals in terms of a restricted form of Vopěnka's Principle. Finally, we establish the equiconsistency of the "$n$-choiceless" extendible cardinals with their original counterparts, and study the consistency strength of other relevant theories.