THE ORIGIN OF SET THEORY

If one is interested in how best to formulate and motivate axioms for set theory, it is worthwhile to take another look at the early history of the subject, right back to the work of its founder, Cantor. Cantor's definition of a set was “any collection into a whole” of “determinate, welldistinguished” sensible or intelligible objects. According to a well-known quip of van Heijenoort, this definition has had as much to do with the subsequent development of set theory as Euclid's definition of point – “that which hath no part” – had to do with the subsequent development of geometry. But in fact the notion of a many made into a one, which is what Cantor's definition makes a set to be, will repay some study.

In order to give concrete meaning to Cantor's abstract definition, our study should begin with a look at the context in which Cantor first felt it desirable or necessary to introduce the notion of set. As is well known, Cantor's general theory of arbitrary sets of arbitrary elements emerged from a previous theory of sets of points on the line or real numbers. This itself emerged from work on Fourier series. The technical details of Cantor's theorems on this topic are irrelevant for present purposes, but the general form of his results should be noted.