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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.10455 |
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Table of Contents:
- 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.