A criterion for uniform finiteness in the imaginary sorts

Abstract

Let T be a theory. If T eliminates $\exists ^\infty$, it need not follow that $T^{\mathrm {eq}}$ eliminates $\exists ^\infty$, as shown by the example of the p-adics. We give a criterion to determine whether $T^{\mathrm {eq}}$ eliminates $\exists ^\infty$. Specifically, we show that $T^{\mathrm {eq}}$ eliminates $\exists ^\infty$ if and only if $\exists ^\infty$ is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of $T^{\mathrm {eq}}$ is unknown. As an application, we show that $T^{\mathrm {eq}}$ eliminates $\exists ^\infty$ when T is a C-minimal expansion of ACVF.