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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2510.18490 |
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| _version_ | 1866917029499371520 |
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| author | Chong, Chi Tat Wong, Tin Lok |
| author_facet | Chong, Chi Tat Wong, Tin Lok |
| contents | Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$Σ^0_2$-bounding}$ in which $Σ^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle Ramsey's Theorem for Pairs, the Cohesive Set Theorem or the Tree Theorem, then there is a $Δ^0_1(A)$-instance of the principle with no solution in $\mathfrak M$ that is arithmetically definable relative to $A$; and (ii) any set of minimal Turing degree in $\mathfrak M$ that is arithmetically definable relative to $A$ has Turing jump equivalent to $A'$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Definability over $\mathrm BΣ^0_2$-models Chong, Chi Tat Wong, Tin Lok Logic 03D55 (Primary), 03C62 (Secondary) Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$Σ^0_2$-bounding}$ in which $Σ^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle Ramsey's Theorem for Pairs, the Cohesive Set Theorem or the Tree Theorem, then there is a $Δ^0_1(A)$-instance of the principle with no solution in $\mathfrak M$ that is arithmetically definable relative to $A$; and (ii) any set of minimal Turing degree in $\mathfrak M$ that is arithmetically definable relative to $A$ has Turing jump equivalent to $A'$. |
| title | Definability over $\mathrm BΣ^0_2$-models |
| topic | Logic 03D55 (Primary), 03C62 (Secondary) |
| url | https://arxiv.org/abs/2510.18490 |