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On collection schemes and Gaifman's splitting theorem (2402.09255v2)

Published 14 Feb 2024 in math.LO

Abstract: 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 $\Sigma_{n+1}$ formulas. 2. For any $K, N \models \mathbf{PA}-$, if $M \subseteq_{\mathrm{cof}} K$, $M \prec_{\Delta_0} K$ and $M \prec N$, then $M \prec_{\Sigma_{n+2}} K$ and $\sup_N(M) \prec_{\Sigma_n} N$. 3. For any $N \models \mathbf{PA}-$, if $M \prec N$, then $M \prec_{\Sigma_{n+2}} \sup_N(M) \prec_{\Sigma_{n}} N$. 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.

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