Computable embeddings and strongly minimal theories

Abstract

Here we prove that if $T$ and $T'$ are strongly minimal theories, where $T'$ satisfies a certain property related to triviality and $T$ does not, and $T'$ is model complete, then there is no computable embedding of $Mod(T)$ into $Mod(T')$. Using this, we answer a question from [4], showing that there is no computable embedding of $VS$ into $ZS$, where $VS is the class of infinite vector spaces over $\mathbb{Q}$, and $ZS$ is the class of models of $Th(\mathbb{Z},S)$. Similarly, we show that there is no computable embedding of $ACF$ into $ZS$, where $ACF$ is the class of algebraically closed fields of characteristic 0.