Reducing First-order Logic to Df₃, Free Algebras

Abstract

Alfred Tarski in 1953 formalized set theory in the equational theory of relation algebras [Tar,53a, Tar,53b]. Why did he do so? Because the equational theory of relation algebras (RA) corresponds to a logic without individual variables, in other words, to a propositional logic. This is why the title of the book [Tar-Giv,87] is “Formalizing set theory without variables”. Tarski got the surprising result that a propositional logic can be strong enough to “express all of mathematics”, to be the arena for mathematics. The classical view before this result was that propositional logics in general were weak in expressive power, decidable, uninteresting in a sense. By using the fact that set theory can be built up in it, Tarski proved that the equational theory of RA is undecidable. This was the first propositional logic shown to be undecidable.