Decidable Models of Ehrenfeucht Theories

We study countable models of Ehrenfeucht theories, i.e., complete theories with a finite number of countable models, strictly larger than 1. The notion of a primely generated model is introduced. It is proved that if all complete types of an Ehrenfeucht theory have arithmetic complexity, then any of the primely generated models of the theory possesses an arithmetically complex isomorphic presentation.