Strongly Minimal Theories with Recursive Models

We give effectiveness conditions on a strongly minimal theory T guaranteeing that all models have computable copies. In particular, we show that if T is strongly minimal and for n ě 1, T X Dn`2 is ∆n, uniformly in n, then every model has a computable copy. Relativizing, we answer a long-standing question in computable model theory, showing that if a strongly minimal theory has a computable model, then every model is arithmetical; in fact, every model has a ∆4 copy.