On collection schemes and Gaifman's splitting theorem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913485956317184 |
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| author | Kurahashi, Taishi Minami, Yoshiaki |
| author_facet | Kurahashi, Taishi Minami, Yoshiaki |
| contents | We study model theoretic characterizations of various collection schemes over $\mathbf{PA}^-$ from the viewpoint of Gaifman's splitting theorem. Among other things, we prove that for any $n \geq 0$ and $M \models \mathbf{PA}^-$, the following are equivalent:
1. $M$ satisfies the collection scheme for $Σ_{n+1}$ formulas.
2. For any $K, N \models \mathbf{PA}^-$, if $M \subseteq_{\mathrm{cof}} K$, $M \prec_{Δ_0} K$ and $M \prec N$, then $M \prec_{Σ_{n+2}} K$ and $\sup_N(M) \prec_{Σ_n} N$.
3. For any $N \models \mathbf{PA}^-$, if $M \prec N$, then $M \prec_{Σ_{n+2}} \sup_N(M) \prec_{Σ_{n}} N$.
Here, $\sup_N(M)$ is the unique $K$ satisfying $M \subseteq_{\mathrm{cof}} K \subseteq_{\mathrm{end}} N$. We also investigate strong collection schemes and parameter-free collection schemes from the similar perspective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On collection schemes and Gaifman's splitting theorem Kurahashi, Taishi Minami, Yoshiaki Logic We study model theoretic characterizations of various collection schemes over $\mathbf{PA}^-$ from the viewpoint of Gaifman's splitting theorem. Among other things, we prove that for any $n \geq 0$ and $M \models \mathbf{PA}^-$, the following are equivalent: 1. $M$ satisfies the collection scheme for $Σ_{n+1}$ formulas. 2. For any $K, N \models \mathbf{PA}^-$, if $M \subseteq_{\mathrm{cof}} K$, $M \prec_{Δ_0} K$ and $M \prec N$, then $M \prec_{Σ_{n+2}} K$ and $\sup_N(M) \prec_{Σ_n} N$. 3. For any $N \models \mathbf{PA}^-$, if $M \prec N$, then $M \prec_{Σ_{n+2}} \sup_N(M) \prec_{Σ_{n}} N$. Here, $\sup_N(M)$ is the unique $K$ satisfying $M \subseteq_{\mathrm{cof}} K \subseteq_{\mathrm{end}} N$. We also investigate strong collection schemes and parameter-free collection schemes from the similar perspective. |
| title | On collection schemes and Gaifman's splitting theorem |
| topic | Logic |
| url | https://arxiv.org/abs/2402.09255 |