Representing all Cylindric Algebras by Twisting on a Problem of Henkin

Abstract

Had the class of cylindric set algebras turned out to be finitely (and/or nicely) axiomatizable, algebraic logic would have evolved along a markedly different path than it did in the past 40 years. Among other things, it would have probably spelled the end of the “abstract” class CAα as a separate subject of research; after all, why bother with abstract algebras, if a few nice extra equations can get us from there to concrete algebras of relations. As it is, Monk’s 1969 result (and its various improvements by algebraic logicians from Andréka to Venema), stating that for α > 2, RCAα is not axiomatizable finitely, meant, among other things, that CAα was here to stay, and its “distance” from RCAα became an important research topic. To mention just one example of how this distance can be measured, we refer to the representation theory of CAs, where sufficient conditions for a CA to be representable are sought. This line of research is typical: the question one asks here is “what is missing from CAs to be representable?”. But Henkin (himself a prolific contributor to representation theory) turned around this question and instead of asking how much CAs needed to be representable, he asked how much set algebras needed to be “distorted” to provide representations for all CAs. Now, if the answer is “very much”, then we haven’t got closer to understanding either CAs or the possible causes of their non-representability.