Theory of Abelian groups with constructive models (Q579238)

In the theory of constructive models, the problem of the existence of a constructible model in a given theory is one of the basic problems. In this paper we obtain a criterion for the existence of a constructive model in a complete theory of Abelian groups. It is shown that any recursively enumerable theory of torsion-free Abelian groups has a constructive model. We construct a recursively enumerable theory of Abelian groups which does not have a constructive model. Problems of (strong) constructibility of prime and saturated models of the theory of Abelian groups are considered as well.