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Berkeley Cardinals and Vop\v{e}nka's Principle
- Publication Year :
- 2024
-
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.<br />Comment: 24 pages
- Subjects :
- Mathematics - Logic
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2404.10455
- Document Type :
- Working Paper