Fix . Let denote the class of cylindric algebras of dimension , and let denote the variety of representable ’s. Let denote first-order logic restricted to the first variables. Roughly, , an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of , namely, its proof theory, while algebraically and geometrically represents the Tarskian semantics of . Unlike Boolean algebras having a Stone representation theorem, . Using combinatorial game theory, we show that the existence of certain finite relation algebras , which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for . Using special cases of such finite ’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on and we re-prove Hirsch and Hodkinson’s result that the class of completely representable ’s is not first-order definable. We show that if is an countable theory that admits elimination of quantifiers, is a cardinal , and is a family of complete nonprincipal types, then can be omitted in an ordinary countable model of .
"Existence of Certain Finite Relation Algebras Implies Failure of Omitting Types for ." Notre Dame J. Formal Logic 61 (4) 503 - 519, November 2020. https://doi.org/10.1215/00294527-2020-0022