Countable Models of Stable Theories

The notion of a normal theory is introduced, and it is proved that for such a theory T, /(N„, 7") = 1 or 3e S(l. Wc also include a short proof of Lachlan's theorem that for superstable 7". /(X0, 7") = 1 or > N(1. (The property of normality is stronger than stability but incomparable to superstability.) 1. Preliminaries. We consider here theories T which are countable and complete (and of course finitary, first order). A long standing question is whether for stable, not N„-categorical T, T must always have infinitely many countable models (up to isomorphism). This was answered positively by Lachlan [1] for T superstable, and other proofs in this case have been given by Lascar [3] and Shelah [7], Here we consider theories which satisfy, in effect, a stronger combinatorial property than "not having the order property", and answer the question positively for such theories, strengthening results in Pillay [6]. I wish to thank the referee for his suggestions, especially concerning the proof of Lemma 4 and an error in our earlier formulation of Lemma 6. Definition 1. (i) The ¿-formula a(x, y) is 7-normal (or just normal) if for any a, b in a model M of T the definable subsets a(x, a) and a(x, b) are either disjoint or equal. (ii) T is normal if every ¿-formula is equivalent modulo F to a Boolean combination of normal formulae (i.e. formulae that are normal no matter how we separate the variables). (Our use of the word normal is motivated by the normal formulas of Lachlan [2], but our notion is of course much cruder.) It is easy to see that a normal theory is stable, either by counting types (if a(x. y) is normal then there can be at most À a-types over any model of power À ) or by observing that a normal formula does not have the order property and thus neither does any Boolean combination of such formulae. Clearly, any theory of equivalence relations El which has elimination of quantifiers is normal, and thus there are normal theories which are co-stable, strictly superstable and strictly stable. Any theory of /\-modules is normal, as a consequence of Baur's "p.p elimination of quantifiers" for modules, and the fact that in an /\-module M, if tp(x, y) is a p.p (positive primitive) formula, then for any äEM <p(x, ä) defines (if consistent) a coset of the subgroup Received by the editors November 30. 1982. Part of the contents of this paper was presented at the 797th meeting of the A. M. S. at the University of Maryland. October 1982. 1980 Mathematics Subject Classification. Primary 03C45: Secondary 03C15.