Completeness theorems for modal logic in second-order arithmetic
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912255953600512 |
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| author | Shimomichi, Sho Takeda, Yuto Yokoyama, Keita |
| author_facet | Shimomichi, Sho Takeda, Yuto Yokoyama, Keita |
| contents | This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in $\mathrm{RCA}_0$, and that, over $\mathrm{RCA}_0$, $\mathrm{ACA}_0$ is equivalent to the strong completeness theorem for modal propositional logic using canonical models. We also consider a simpler version of the strong completeness theorem without referring to canonical models and show that it is equivalent to $\mathrm{WKL}_0$ over $\mathrm{RCA}_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completeness theorems for modal logic in second-order arithmetic Shimomichi, Sho Takeda, Yuto Yokoyama, Keita Logic This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in $\mathrm{RCA}_0$, and that, over $\mathrm{RCA}_0$, $\mathrm{ACA}_0$ is equivalent to the strong completeness theorem for modal propositional logic using canonical models. We also consider a simpler version of the strong completeness theorem without referring to canonical models and show that it is equivalent to $\mathrm{WKL}_0$ over $\mathrm{RCA}_0$. |
| title | Completeness theorems for modal logic in second-order arithmetic |
| topic | Logic |
| url | https://arxiv.org/abs/2503.01191 |