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## The beginnings of set theory

While the apparent paradoxes associated with infinite sets have been known since the Renaissance, they did not receive serious attention until the nineteenth century, when Bolzano made a more systematic study of them in [1]. The issue arose again when progress in the development of analysis demanded a rigorous definition of the real numbers. Increased standards of rigor and the theory of functions of several variables necessitated a complete arithmetization of the real numbers. In order to improve upon Cauchy's still partly geometric arguments for many of the central theorems in analysis, Dedekind and Cantor, both students of Weierstrass, gave two (equivalent) definitions of the real numbers not employing any geometric concepts.

Cantor's definition of the real numbers [11, p. 577,] is based on the concept of a Cauchy sequence, a notion which Cauchy had used to give an ``internal'' criterion for a sequence of numbers to converge, and one which makes no reference to its limit. Once Cantor had a suitable definition for the real numbers, he was in a position to study them as an infinite set.

Bolzano had made it clear in [1] that he considered the property of a one-to-one correspondence between an infinite set and a proper subset fundamental to the nature of infinite sets. After Cantor realized that this property should be used as the very definition of ``infinite set'', it was an easy task for him to demonstrate both the countability of the rational numbers [3, pp. 110--111,] (using a nonstandard order relation on the rationals equivalent to the usual diagonal argument) as well as the uncountability of the real numbers [11, pp. 579--580,]. The latter proof can of course immediately be generalized to prove that the power set operation increases cardinality, thus providing the basis for Cantor's system of infinite numbers. Cantor's continuum hypothesis [11, pp. 580--581,] (which he considered to be a theorem) became one of the important modern problems in set theory, which was solved only relatively recently.

Next: Solutions of algebraic Up: Great Problems of Mathematics: Previous: Area and the

D. Pengelley and R. Laubenbacher
Sun Feb 7 00:21:52 MST 1999