Martin's axiom, omitting types, and complete representations in algebraic logic
An \(n\)-dimensional (\(n\)-finite) cylindric algebra \(A\) is said to be completely representable if, for every non-zero element \(a \in A\), there is a completely additive homomorphism \(f\) from \(A\) to an \(n\)-dimensional cylindric set algebra with \(f(a) \neq 0\). One of author's results gives a new characterisation of countable completely representable finite-dimensional cylindric (and relational) algebras via special neat embeddings. It is also shown that a certain statement concerning the existence of representations for cylindric algebras is equivalent to a certain form of Martin's axiom (restricted to countable Boolean algebras) and turns out to be connected also with omitting types for \(L_{\omega\omega}\). As an application, Henkin and Orey's omitting types theorem is shown to be false for first-order logics with a finite number of variables. Some open problems are posed, and recent results by H. Andréka and by R. Hirsch related to two of them are discussed.
scientific article; zbMATH DE number 1880460