Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916487268139008 |
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| author | Hamkins, Joel David Yao, Bokai |
| author_facet | Hamkins, Joel David Yao, Bokai |
| contents | After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $κ$ is supercompact if and only if every $Π^1_1$ sentence true in a structure $M$ (of any size) containing $κ$ in a language of size less than $κ$ is also true in a substructure $m\prec M$ of size less than $κ$ with $m\capκ\inκ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_09766 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal Hamkins, Joel David Yao, Bokai Logic 03E30, 03E55, 03E65 After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $κ$ is supercompact if and only if every $Π^1_1$ sentence true in a structure $M$ (of any size) containing $κ$ in a language of size less than $κ$ is also true in a substructure $m\prec M$ of size less than $κ$ with $m\capκ\inκ$. |
| title | Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal |
| topic | Logic 03E30, 03E55, 03E65 |
| url | https://arxiv.org/abs/2204.09766 |