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The classification of small weakly minimal sets. II

Abstract The main result is Vaught's conjecture for weakly minimal, locally modular and non-ω-stable theories. The more general results yielding this are the following. Theorem A. Suppose that T is a small unidimensional theory and D is a weakly minimal set, definable over the finite set B. Then for all finite A ⊂ D there are only finitely many nonalgebraic strong types over B realized in acl(A) ∩ D. Theorem B. Suppose that T is a small, unidimensional, non-ω-stable theory such that the universe is weakly minimal and locally modular. Then for all finite A there is a finite B ⊂ cl(A) such that a ∈ cl(A) iff a ∈ cl(b) for some b ∈ B. Recall the property (S) defined in the abstract of [B1]. Theorem C. Let T be as in Theorem B. Then, if T does not satisfy (S), T hasmany countable models. Combining Theorem C and the results in [B1] we obtain Vaught's conjecture for such theories.