On the finitization problem of algebraic logic
by
Gábor Sági
Technical University, Budapest

The Finitization Problem is one of the oldest problems of Algebraic Logic and today it is not only a problem, but a research direction. At the logical side, the Finitization Problem asks whether is it possible to extend or modify the usual first order logic in order to obtain a new logic L in which one can postulate a finite set \Sigma of valid formula schemas such that all the valid formula schemas of L can be derived from \Sigma? This problem is equivalent with finding a finitely axiomatizable class of algebras of relations with higher rank. Moreover, the Finitization Problem has a purely algebraic character: it can be regarded as defining a ``complicated enough'' discriminator (or congruence distributive) variety V and developing a representation theory for V.

We will present some recent results and further questions about the Finitization Problem.

Date received: August 4, 1999