Berkeley Cardinals and Vopenka's Principle

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 “C (n) -Cardinals” and of Bagaria and Poveda’s work in “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 determine the consistency strength of some relevant theories that arise.