Strongly minimal theories with recursive models

Abstract

We give effectiveness conditions on a strongly minimal theory $T$ guaranteeing that all countable models have computable copies. In particular, we show that if $T$ is strongly minimal and for all $n\ge 1$, $T\cap {\exists }_{n+2}$ is ${\Delta }_n^0$, uniformly in $n$, then every countable model has a computable copy. A longstanding question of computable model theory asked whether for a strongly minimal theory with one computable model, every countable model has an arithmetical copy. Relativizing our main result, we get the fact that if there is one computable model, then every countable model has a ${\Delta }_4^0$ copy.