set theory

The modern theory of sets was largely inspired by Cantor, whose proof that the set of real numbers could not be put into a one-to-one correspondence with the set of natural numbers opened the door to the set-theoretic hierarchy, and to the study of transfinite numbers. The first proper axiomatization of the theory was that of Ernst Zermelo (1871–1953) in 1908. The axiomatization followed intense controversy over the nature of the set-theoretic hierarchy, the legitimacy of the axiom of choice, and the right approach to the paradoxes lying at the centre of naïve views about sets, of which the best known is Russell's paradox . Classical set theory uses the axiomatization of Zermelo, augmented by the axiom of replacement due to A. Fraenkel (1891–1965). It has been shown that this is equivalent to a natural ‘iterative’ conception, whereby starting with the empty set or null set, and forming only sets of sets, the entire set-theoretic hierarchy can be generated. Philosophically set theory is central because sets are the purest mathematical objects, and it is known that the rest of mathematics can be formulated within set theory (so that numbers, relations, and functions all become particular sets). Particular topics within set theory are indexed under their own headings.