On Vaught’s Conjecture and finitely valued MV algebras

Abstract

We show that the complete first order theory of an MV algebra has \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$2^{\aleph _0}$\end{document} 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$2^{\aleph _0}$\end{document} and that all ω‐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 ω‐categorical.