Some variants of Vaught's conjecture from the perspective of algebraic logic
Gábor Sági, Dorottya Sziráki · Logic Journal of IGPL · 2012
Vaught's Conjecture states that if Σ is a complete first order theory in a countable language such that Σ has uncountably many pairwise non-isomorphic countably infinite models, then Σ has 2ℵ0 many pairwise non-isomorphic countably infinite models. Continuing investigations initiated in Sági (2011, submitted for publication), we apply methods of algebraic logic to study some variants of Vaught's conjecture. More concretely, let S ⊆ ωω be a σ-compact monoid. We prove, among other things, that if a complete first order theory Σ has at least ℵ1 many countable models that cannot be elementarily embedded into each other by elements of S, then, in fact, Σ has continuum many such models. We also study-related questions in the context of equality free logics and obtain similar results. Our proofs are based on the representation theory of cylindric and quasi-polyadic algebras (for details see Henkin, Monk and Tarski (cylindric Algebras Part 1 and Part 2)) and topological properties of the Stone spaces of these algebras.