Abstract
Fix 2 < n < omega. 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) not subset of 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, lambda is a cardinal < 2(N0), and F = <Gamma(i) : i < lambda > is a family of complete nonprincipal types, then F can be omitted in an ordinary countable model of T.