Recursive Models of Theories with Few Models

We begin by presenting some basic definitions from effective model theory. A recursive structure is one with a recursive domain and uniformly recursive atomic relations. Without lost of generality, we can always suppose that the domain of every recursive structure is the set of all naturall numbers ω and that its language does not contain function symbols. If a structure A is isomorphic to a recursive structure B, then A is recursively presentable and B is a recursive presentation of A. Let σ be an effective signature. Let σ0 ⊂ σ1 ⊂ σ2 ⊂ . . . be an effective sequence of finite signatures such that σ = ⋃ t σt. It is clear that a structure A of signature σ is recursive if and only if there exists an effective sequence A0 ⊂ A1 ⊂ A2 ⊂ . . . of finite structures such that for each i the domain of Ai is {0, . . . , ti}, the function i→ ti is recursive, Ai is a structure of signature σi, Ai+1 is an expansion and ∗Partially supported by ARO through MSI, Cornell University, DAAL03-91-C0027. †Partially supported by MSI, Cornell University and the NSF grant DMS 9500983 §Partially supported by NSF Grant DMS-9204308, DMS-9503503 and ARO through MSI, Cornell University, DAAL-03-C-0027.