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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2411.16338 |
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| _version_ | 1866912132144037888 |
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| author | Hashimoto, Koki |
| author_facet | Hashimoto, Koki |
| contents | This paper presents two types of results related to hyperarithmetic analysis. First, we introduce new variants of the dependent choice axiom, namely $\mathrm{unique}~Π^1_0(\mathrm{resp.}~Σ^1_1)\text{-}\mathsf{DC}_0$ and $\mathrm{finite}~Π^1_0(\mathrm{resp.}~Σ^1_1)\text{-}\mathsf{DC}_0$. These variants imply $\mathsf{ACA}_0^+$ but do not imply $Σ^1_1\mathrm{~Induction}$. We also demonstrate that these variants belong to hyperarithmetic analysis and explore their implications with well-known theories in hyperarithmetic analysis. Second, we show that $\mathsf{RFN}^{-1}(\mathsf{ATR}_0)$, a class of theories defined using the $ω$-model reflection axiom, approximates to some extent hyperarithmetic analysis, and investigate the similarities between this class and hyperarithmetic analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16338 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of hyperarithmetic analysis by $ω$-model reflection Hashimoto, Koki Logic This paper presents two types of results related to hyperarithmetic analysis. First, we introduce new variants of the dependent choice axiom, namely $\mathrm{unique}~Π^1_0(\mathrm{resp.}~Σ^1_1)\text{-}\mathsf{DC}_0$ and $\mathrm{finite}~Π^1_0(\mathrm{resp.}~Σ^1_1)\text{-}\mathsf{DC}_0$. These variants imply $\mathsf{ACA}_0^+$ but do not imply $Σ^1_1\mathrm{~Induction}$. We also demonstrate that these variants belong to hyperarithmetic analysis and explore their implications with well-known theories in hyperarithmetic analysis. Second, we show that $\mathsf{RFN}^{-1}(\mathsf{ATR}_0)$, a class of theories defined using the $ω$-model reflection axiom, approximates to some extent hyperarithmetic analysis, and investigate the similarities between this class and hyperarithmetic analysis. |
| title | Approximation of hyperarithmetic analysis by $ω$-model reflection |
| topic | Logic |
| url | https://arxiv.org/abs/2411.16338 |