Degrees of recursively saturated models

Abstract

Using relativizations of results of Goncharov and Peretyat’kin on decidable homogeneous models, we prove that if M M is S S -saturated for some Scott set S S , and F F is an enumeration of S S , then M M has a presentation recursive in F 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).