Some variants of Vaught’s conjecture from the perspective of algebraic logic

Sági, Gábor and Sziráki, Dorottya (2012) Some variants of Vaught’s conjecture from the perspective of algebraic logic. Logic Journal of IGPL, 20 (6). pp. 1064-1082.

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Abstract

Vaught’s Conjecture states that if T is a complete first order theory in a countable language such that T has uncountably many pairwise non-isomorphic countably infinite models, then T has 2^ℵ_0 many pairwise non-isomorphic countably infinite models. Continuing investigations initiated in S´agi, we apply methods of algebraic logic to study some variants of Vaught’s conjecture. More concretely, let S be a σ-compact monoid of selfmaps of the the natural numbers. We prove, among other things, that if a complete first order theory T has at least ℵ1 many countable models that cannot be elementarily embedded into each other by elements of S, then, in fact, T 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.

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