Axiom $\mathcal{A}$ and supercompactness
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914839326097408 |
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| author | Poveda, Alejandro |
| author_facet | Poveda, Alejandro |
| contents | We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \cite{HMP} and provides a definitive answer to a question of Bagaria \cite[p.19]{Bag}. This configuration is a consequence of a new axiom we introduce -- called $\mathcal{A}$ -- which is showed to be compatible with Woodin's $I_0$ cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12776 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Axiom $\mathcal{A}$ and supercompactness Poveda, Alejandro Logic We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \cite{HMP} and provides a definitive answer to a question of Bagaria \cite[p.19]{Bag}. This configuration is a consequence of a new axiom we introduce -- called $\mathcal{A}$ -- which is showed to be compatible with Woodin's $I_0$ cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing. |
| title | Axiom $\mathcal{A}$ and supercompactness |
| topic | Logic |
| url | https://arxiv.org/abs/2406.12776 |