Tame Theories with Hyperarithmetic Homogeneous Models

A tame theory is a decidable first-order theory with only countably many countable models, and all complete types recursive. It is shown here that the recursive complexity of countable homogeneous models of tame theories is unbounded in the hyperarithmetic hierarchy. Recursive model theory considers model-theoretic objects and constructions from the point of view of recursion-theoretic complexity. For instance, must isomorphism types of countable models that are structurally simple have models of low recursion-theoretic complexity? This paper considers one such question involving countable homogeneous models. If £ is a countable homogeneous model of a complete theory T, then is £ necessarily decidable in the degree of Tl Since there are complete theories with 2m complete types, the answer is obviously "no". On the other hand, by results in [1], if T has only finitely many countable models and all of the complete types of the theory are recursive, then the answer is "yes". This paper considers decidable theories T that have at most countably many countable models and only recursive complete types. Call such a theory tame. The notational conventions in the paper are standard. Fact. Every countable homogeneous model of a tame theory T is hyperarithmetic. Proof. Let {p^i < co} be an effective enumeration of all the partial recursive functions ß: N —► TV, and let {B(\i < co} be an effective enumeration of all formulas in L(T). Since all of the types of T are recursive, we can fix a hyperarithmetic set 7 C TV such that (i) Vi € I[{0j\ßi(j) = 1Í7 < <y} is a complete type of T]; and (ii) vr complete type of T miel Vj[8j G T iff ß.(J) = I]. For a countable model il IT, say that W CI is a witness set for il iff there is some enumeration {jS.\i < co} of |il|<cu such that (1) V«e W [ßn is total]; Received by the editors September 15, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03C57. Research for this paper was partially supported by Australian Research Grant #8215113 and NSF Grant #DMS 8501521. © 1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page