Effective model theory vs. recursive model theory

Abstract

Recursive model theory is supposed to be the study of the effectiveness of constructions and theorems in model theory. This often involves getting “effective” versions of various classical model-theoretic notions. The traditional way of doing this is to restrict attention to recursive models, and recursive isomorphisms between them, etc. Thus for example the following definition appears in the literature (in [3] and [1]).Definition. Given a recursive model A and an n Є ω, a subset R ⊆ An is called intrinsically r.e. provided that for every recursive model B ≈ A, the isomorphic image in B of R is an r.e. subset of Bn.It is clear that if R is definable by a (recursive, infinitary) Σ10 formula (with finitely many parameters from A), then R is intrinsically r.e. It seems natural for the converse to be true. Indeed, provided that (A, R) is sufficiently “regular” in a sense made precise in a theorem of Ash and Nerode (see [3]), the converse is true. However, if we drop the (rather strong) regularity conditions, there exist “pathological” examples of intrinsically r.e. relations which are not definable by a Σ10 formula (see [7]).In this paper, we suggest a rather different approach to studying the effectiveness of model theory, an approach we have dubbed “effective model theory”. The basic idea is to allow arbitrary nonrecursive models, but to require all notions to be relativized to the complexity of the models involved. (Much the same notion has been used in [2] under the name “relatively recursive model theory”.) Thus for example we have the following effective model theory version of the property of being intrinsically r.e.