An isomorphism theorem for models of Weak K\'onig's Lemma without primitive recursion
| Autor: | Fiori-Carones, Marta, Kołodziejczyk, Leszek Aleksander, Wong, Tin Lok, Yokoyama, Keita |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | We prove that if $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are countable models of the theory $\mathrm{WKL}^*_0$ such that $\mathrm{I}\Sigma_1(A)$ fails for some $A \in \mathcal{X} \cap \mathcal{Y}$, then $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are isomorphic. As a consequence, the analytic hierarchy collapses to $\Delta^1_1$ provably in $\mathrm{WKL}^*_0 + \neg\mathrm{I}\Sigma^0_1$, and $\mathrm{WKL}$ is the strongest $\Pi^1_2$ statement that is $\Pi^1_1$-conservative over $\mathrm{RCA}^*_0 + \neg\mathrm{I}\Sigma^0_1$. Applying our results to the $\Delta^0_n$-definable sets in models of $\mathrm{RCA}^*_0 + \mathrm{B}\Sigma^0_n + \neg\mathrm{I}\Sigma^0_n$ that also satisfy an appropriate relativization of Weak K\"onig's Lemma, we prove that for each $n \ge 1$, the set of $\Pi^1_2$ sentences that are $\Pi^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}\Sigma^0_n + \neg\mathrm{I}\Sigma^0_n$ is c.e. In contrast, we prove that the set of $\Pi^1_2$ sentences that are $\Pi^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}\Sigma^0_n$ is $\Pi_2$-complete. This answers a question of Towsner. We also show that $\mathrm{RCA}_0 + \mathrm{RT}^2_2$ is $\Pi^1_1$-conservative over $\mathrm{B}\Sigma^0_2$ if and only if it is conservative over $\mathrm{B}\Sigma^0_2$ with respect to $\forall \Pi^0_5$ sentences. Comment: 29 pages. Somewhat more polished version compared to v1, with small improvements and simplifications throughout the text but no major mathematical changes. Introduction slightly expanded to point out model-theoretic aspects of the paper |
| Databáze: | arXiv |
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