Foundations

Abstract

The year of birth of set theory can be regarded as 1872. In this year, Georg Cantor introduced the notion of a transfinite (infinite) ordinal number His investigations, which soon made set theory an independent area of mathematics, met with mistrust and sometimes open hostility from other mathematicians. At the start of the last decade of the century, set theory came into vogue and was being applied more and more extensively to analysis and geometry. Just when Cantor had found high recognition, and when he had put the finishing touches to his investigations, he himself discovered the first antinomy in his system, which was also found two years later, in 1897, by Burali-Forti. Neither of them could offer any solution, but they did not take the problem too seriously, since it arose in a pretty technical area, namely, the theory of well-ordered sets. They believed that a revision of proofs and theorems could settle the situation, as was often the case earlier, under similar circumstances. These hopes were radically dashed when, in 1902, Bertrand Russell published his famous antinomy which occurred just at the beginning of set theoretical investigations and demonstrated that something wasn’t right in the foundations of that discipline. The disclosure of that antinomy led to a foundational crisis in mathematics. Several ways out of this crisis have been developed, the most successful one being without any doubt the axiomatic method. And amongst the various axiomatic systems of set theory, the system which was begun by Zermelo and completed by Fraenkel and Skolem is, adding the axiom of choice, perhaps the most known and the most practical one. The results of our book are set forth in this theory, denoted by ZFC, the axioms of which are represented below. We always begin with a naive formulation of the particular axiom of ZFC and will then make precise the notion of a “property” occurring in two axioms.