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

Abstract

Fix 2 < n < ω . Let CA n denote the class of cylindric algebras of dimension n , and let RCA n denote the variety of representable CA n ’s. Let L n denote first-order logic restricted to the first n variables. Roughly, CA n , an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of L n , namely, its proof theory, while RCA n algebraically and geometrically represents the Tarskian semantics of L n . Unlike Boolean algebras having a Stone representation theorem, RCA n ⊊ CA n . Using combinatorial game theory, we show that the existence of certain finite relation algebras RA , which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for L n . Using special cases of such finite RA ’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on RCA n and we re-prove Hirsch and Hodkinson’s result that the class of completely representable CA n ’s is not first-order definable. We show that if T is an L n countable theory that admits elimination of quantifiers, λ is a cardinal < 2 ℵ 0 , and F = 〈 Γ i : i < λ 〉 is a family of complete nonprincipal types, then F can be omitted in an ordinary countable model of T .