Berkeley Cardinals and Vopěnka's Principle
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910924512690176 |
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| author | Mohammd, Marwan Salam |
| author_facet | Mohammd, Marwan Salam |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Berkeley Cardinals and Vopěnka's Principle Mohammd, Marwan Salam Logic 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. |
| title | Berkeley Cardinals and Vopěnka's Principle |
| topic | Logic |
| url | https://arxiv.org/abs/2404.10455 |