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 $T^{eq}$ eliminates ${\exists }^{\infty }$, as shown by the example of the $p$-adics. We give a criterion to determine whether $T^{eq}$ eliminates ${\exists }^{\infty }$. Specifically, we show that $T^{eq}$ 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 $T^{eq}$ is unknown. As an application, we show that $T^{eq}$ eliminates ${\exists }^{\infty }$ when $T$ is a C-minimal expansion of ACVF.

6 pages