The Genesis and Development of Set Theory

Abstract

U, If one year can be specified as the time when set theory started, that year should probably be 1874, the year in which Georg Cantor's paper [2] was published establishing the countability of the set of real algebraic numbers and the noncountability of the set of real numbers. The proof that the set of real numbers is not countable used nested intervals instead of Cantor's famous diagonal process, which appeared in a later work [9]. According to Fraenkel, Cantor himself had first thought that the continuum could be put in one-to-one correspondence with the set of natural numbers [12, p. 237]. Cantor observed in the introduction of his paper that combining the two theorems gives a result first proved by Liouville-that in each given interval there exist infinitely many transcendental (nonalgebraic) real numbers. There was some difficulty surrounding the publication of Cantor's next article [3] in 1878. The work remained in the publishing room of Crelle's Journal longer than usual for that time, apparently because Cantor's ideas were rejected by Leopold Kronecker, who was on the editing staff of the journal [12, p. 198]. The work was eventually published in the journal, but it was the last article Cantor published there.