An instance of Vaught's conjecture using algebraic logic

Abstract

Let \phi be a first order formula and M be a countable model. \phi^M denotes the set of all assignments that satisfy \phi in M. Let M, N be countable models. A formula \phi distinguishes these models if |\phi^M|\neq |\phi^N|. We show that the number of distinguishable countable models for a complete countable first order theory, satisfies the Glimm Effross dichotomy, hence also Vaught's conjecture. The proof in the presence of equality is fairly easy, though we need infinite conjunctions to implement it and somewhat heavy theorems from descriptive set theory. However, the case without equality seems to be much more difficult (because we cannot count without equality) and in this case algebraic logic comes to our rescue via the representation theory of locally finite quasi polyadic algebras. We show that the equivalence relation induced by such (indistinguishable) models is a Borel subset of the product of the Stone Polish space (consisting of ultarfilters corresponding to models) with itself. We also formuate and prove a Vaught's theorem for certian infinitary extensions of first order logic via the representation theory of dimension complemented algebras. In all case we also count the number of non isomorphic models omitting < covK many non isolated types (the latter is the least cardinal such that the Baire Category theorem fails and also the largest cardinal such tha Martin's axiom restrcited to countable Boolean algebra hold.) In such cases, for locally finite algebras and dimension complemented ones we obtain the same number of non-ismorphic distinguishable models.