On Vaught's Conjecture and finitely valued MV algebras

We show that the complete first order theory of an MV algebra has continuum many countable models unless the MV algebra is finitely valued. So, Vaught's Conjecture holds for all MV algebras except, possibly, for finitely valued ones. Additionally, we show that the complete theories of finitely valued MV algebras are continuum many and that all omega-categorical complete theories of MV algebras are finitely axiomatizable and decidable. As a final result we prove that the free algebra on countably many generators of any locally finite variety of MV algebras is omega-categorical. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim