Strongly minimal theories with recursive models
Uri Andrews, Julia F. Knight · Journal of the European Mathematical Society · 2018
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\geq 1 , T\cap\exists_{n+2} is \Delta^0_n , 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^0_4 copy.