On collection schemes and Gaifman's splitting theorem

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.