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

Abstract

Fix $2<n<\omega$. Let ${\mathsf{CA}}_n$ denote the class of cylindric algebras of dimension $n$, and let ${\mathbf{RCA}}_n$ denote the variety of representable ${\mathsf{CA}}_n$’s. Let $L_n$ denote first-order logic restricted to the first $n$ variables. Roughly, ${\mathsf{CA}}_n$, an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of $L_n$, namely, its proof theory, while ${\mathbf{RCA}}_n$ algebraically and geometrically represents the Tarskian semantics of $L_n$. Unlike Boolean algebras having a Stone representation theorem, ${\mathbf{RCA}}_n⊊{\mathsf{CA}}_n$. Using combinatorial game theory, we show that the existence of certain finite relation algebras $\mathsf{RA}$, 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 $\mathsf{RA}$’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on ${\mathbf{RCA}}_n$ and we re-prove Hirsch and Hodkinson’s result that the class of completely representable ${\mathsf{CA}}_n$’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 $<2^{{\aleph }_0}$, and $\mathbf{F}=\langle {\Gamma }_i:i<\lambda \rangle$ is a family of complete nonprincipal types, then $\mathbf{F}$ can be omitted in an ordinary countable model of $T$.