Existence of Certain Finite Relation Algebras Implies Failure of Omitting Types for L n

Abstract

Fix 2 < n < ω. Let CAn denotes the class of cylindric algebras of dimension n, and RCAn denotes the variety of representable CAns. Let Ln denote rst order logic restricted to the rst n variables. Roughly CAn, an instance of Boolean algbras with operators, is the algebraic counterpart of the syntax of Ln, namely, its proof theory, while RCAn represents algebraically and geometrically Tarskian semantics of Ln. Unlike Boolean algebras, having a Stone representation theorem, RCAn ( CAn. Using combinatorial game theory, we show that the existence of certain nite relation algebras RAs, which are algebras whose domain consists of binary relations, imply that the celebrated Henkin omitting types theorem, fails in a very strong sense for Ln. Using special cases of such nite RAs, we recover the classical nonnite axiomatizability results of Monk, Maddux and Biro on RCAn and we reprove Hirsch and Hodkinson's result that the class of completely representable CAns is not rst order de nable. We show that if T is an Ln countable theory that admits elimination of quanti ers, λ is a cardinal < 2א0 and F = ⟨Γi : i < λ⟩ is a family of complete non-principal types, then F can be omitted in an ordinary countable model of T .