Vaught’s conjecture for superstable theories of finite rank

In [R. Vaught, Denumerable models of complete theories, in: Infinitistic Methods, Pregamon, London, 1961, pp. 303–321] Vaught conjectured that a countable first order theory has countably many or 2ℵ0 many countable models. Here, the following special case is proved. Theorem – If T is a superstable theory of finite rank with <2ℵ0 many countable models, then T has countably many countable models. The basic idea is to associate with a theory a ⋀-definable group G (called the structure group) which controls the isomorphism types of countable models of the theory. The theory of modules is used to show that for M⊧T, G∩M is, essentially, the direct sum of copies of finitely many finitely generated subgroups. This is the principal ingredient in the proof of the following main theorem, from which Vaught’s conjecture follows immediately. Structure Theorem – Let T be a countable superstable theory of finite rank with <2ℵ0 many countable models. Then for M a countable model of Tthere is a finite A⊂M and a J⊂M such that M is prime over A∪J , J is A-independent and is finite.