What the finitization problem is not

Abstract

0. Introduction. The finitization problem is one of the problems considered important in algebraic logic. Since it is often misunderstood, it seems desirable to try to develop a better understanding of what it is and what it is not about. The finitization problem is (at least partly) motivated by the following observations (1)–(3): (1) The natural algebraic counterpart of propositional logic is the variety of Boolean algebras which is axiomatizable by finitely many equations. (2) The natural algebraic counterpart of first-order logic Lωω is the variety RCAω of representable cylindric algebras which is very far from being axiomatizable by a finite schema of equations, cf. [HMT], Theorem 4.1.3, and Andreka [A91]. (3) The algebraic counterpart of Lωω without equality is the variety RQPAω of representable quasi-polyadic algebras which is also not axiomatizable by a finite schema, see Sain–Thompson [ST]. (Similar negative results apply to the finitevariable fragments of Lωω. The algebraic counterparts here are finite-dimensional representable cylindric or polyadic algebras and relation algebras.) The negative results (2) and (3) do have purely logical consequences motivating the (nonalgebraic) logician to look into the question, cf. e.g. §4 of Sain [S87], Simon [S90], [S91], Venema [V90], [V92], Nemeti [N91], but see also the present Appendix A. Observations (2) and (3) above are in contrast with (1). So there seems to be a sharp contrast between the behaviour of propositional logic and quantifier logics. It is natural to ask whether the nonfinitizability in (2) and (3) is an unavoidable