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.