Characterizing Fragments of Collection in Set Theory by Model-Theoretic Properties

Abstract

We compare the model theory of the weak set theory $\mathsf{DB}_0$ with that of the algebraic theory $\mathsf{PA}^-$. Every model of $\mathsf{DB}_0$ has a proper taller* end extension with an exact transitive cover, paralleling the corresponding end-extension result for $\mathsf{PA}^-$. More substantially, we prove that $\mathsf{DB}_0$ and $\mathsf{PA}^-$ are mutually interpretable. The new direction interprets $\mathsf{DB}_0$ in $I\Delta_0+\Omega_1$ by finite rooted acyclic graphs; bisimulation supplies equality, and bounded truth on the graphs supplies $\Delta_0$-Separation. We then characterize fragments of set-theoretic Collection by Gaifman-style splitting and cofinal elementarity. Finally, we separate two end-extension mechanisms. Without a resolution, Kaufmann's construction characterizes the relevant Collection fragments for countable and locally for $\aleph_1$-like models. With a strict transitive resolution, Power Set, Infinity, and strong Collection, a finite-strength Keisler--Morley construction gives $\Sigma_N$-elementary taller* end extensions whose output retains strong $\Sigma_{N-2}$-Collection. The proof is choice-free inside the model and also recovers the classical $\mathsf{ZF}$ theorem.