On the Prime Model Property

Assume T is superstable, Φ(x) is a formula over ∅, Q = Φ(M∗) is countable and KQ = {M : M is countable and Φ(M) = Q}. We investigate models in KQ assuming KQ has the prime model property. We prove some corollaries on the number of models inKQ. We show an example of an ω-stable T and Q with KQ having exactly 3 models. First we fix the general set-up. Throughout, T is a countable complete theory in a first-order language L, Φ(x) is a formula of L without parameters, M∗ is a countable model of T and Q = Φ(M∗). We work within a monster model C = C of T . All models of T we consider are elementary submodels of C. Let KQ = {M : M is countable and Φ(M) = Q} and let I(KQ) be the number of models in KQ, up to isomorphism. The goal of this note is to develop some model theory of KQ. As I pointed out in [Ne1, p.651], in general KQ can not be treated with all common model-theoretic tools, since for example KQ does not necessarily have the joint embedding property. It turns out however that if we assume just that KQ has the prime model property (defined below), then it is much more manageable, at least for stable T . KQ really may differ from an elementary class: I have found an example of KQ with I(KQ) = 2, while a theorem of Vaught [Sa] says that I(T,א0) 6= 2. In this example T is weakly minimal, but KQ does not have the prime model property. In this paper we give an example of an ω-stable T with I(T,א0) = א0 and a strongly minimal Q with I(KQ) = 3. Now we recall some basic notions from [Ne1]. We call f ∈ Aut(C) a Q-mapping if f [Q] = Q. For a ⊂M ∈ KQ let AutQ(C/a) be the group of Q-mappings fixing a pointwise. This group acts in a Borel way on S(Qa), inducing equivalence relation E(a) on S(Qa). The orbits of this action, that is the E(a)-classes, are called pseudotypes over Qa. When a = ∅, we use E to denote E(a). Pseudotypes are Borel subsets of S(Qa), and E(a) is analytic. We say that p ∈ S(Qa) is Q-isolated over a if p/E(a) is not meager. p/E(a) is the pseudotype of p over Qa (that is the equivalence class as a subset of S(Qa), not a quotient of some sort). Q-isolation has many nice properties (see [Ne1, Ne2]), for instance if Q-isolated types are dense in S(Q) then there is M ∈ KQ such that each b ⊂M satisfies a Q-isolated type over Q. Such an M is called Q-atomic. A Q-atomic model is unique up to isomorphism. Also, tp(ab/Q) is Q-isolated iff tp(a/Q) is Q-isolated and tp(b/Qa) is Q-isolated over a. Received by the editors August 26, 1994 and, in revised form, February 13, 1995. 1991 Mathematics Subject Classification. Primary 03C15, 03C45. c ©1996 American Mathematical Society