2024/04/16 by Mohammd, Marwan Salam
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2404.10455
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.