Degrees of Recursively Saturated Models

Using relativizations of results of Goncharov and Peretyat'kin on decidable homogeneous models, we prove that if M is S-saturated for some Scott set S, and F is an enumeration of S, then M has a presentation recursive in F. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii) internally finite initial segments and (iii) nonstandard residue fields. We also use our results to simplify Solovay's characterization of degrees coding nonstandard models of Th(N). 0. Introduction. The classic theorem of Tennenbaum [T] says that there is no recursive nonstandard model of Peano arithmetic. Indeed, if (u, ffi, 0) N P, and (to, ffi, O) is nonstandard, then neither © nor O is recursive. In the above situation, (w, ffi) and («, O) are recursively saturated, and nowadays one formulates the theorem as the statement that there are no recursive, recursively saturated models of either Presburger or Skolem arithmetic. Let us say a theory T has the Tennenbaum property if T has no recursive, recursively saturated models. Macintyre proved in [Macl] that most theories of fields have the Tennenbaum property. In this paper we look at a natural refinement of the Tennenbaum property. For any countable structure M we look at Turing degrees of presentations of M. If M N P, M nonstandard, and M is isomorphic to (w, ffi, O), 0 is not the Turing degree of ffi, nor of O. But given M, what are the possible degrees for ffi, O? And how do these relate to the possible degrees for ( ffi, O )? Here is a sample of results proved below: (a) the set of degrees for ffi equals the set of degrees for O ; (b) the set of degrees for ffi and O are closed upwards; (c) if M is recursively saturated, the set of degrees for ( ffi, O ) is the same as the set of degrees for ffi. These are corollaries of a very general theorem about recursively saturated models of effectively perfect theories T. If 2Í è T, 31 recursively saturated, there is associated with 21 a canonical Scott set S. Our main theorem relates degrees of presentations of 21 and degrees of presentations of S, showing that the sets are mutually dense. The essential tool we use is the Goncharov-Peretyat'kin Theorem on homogeneous models [G, P], suitably relativized. A priority argument is needed here. Received by the editors November 4, 1982. 1980 Mathematics Subject Classification. Primary 03H15; Secondary 03D45, 03C50.