Berkeley Cardinals and Vop\v{e}nka's Principle

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 Vop\v{e}nka'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 Vop\v{e}nka'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.