set theory

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.

Philosophy dictionary. . 2011.

Look at other dictionaries:

  • Set theory — set theory …   Philosophy dictionary

  • set theory — n. the branch of mathematics that deals with the properties and relations of sets: see SET (n. 7) …   English World dictionary

  • set theory — ► NOUN ▪ the branch of mathematics concerned with the formal properties and applications of sets …   English terms dictionary

  • Set theory — This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects …   Wikipedia

  • set theory — the branch of mathematics that deals with relations between sets. [1940 45] * * * Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its… …   Universalium

  • set theory — set′ the ory n. math. the branch of mathematics that deals with relations between sets • Etymology: 1940–45 …   From formal English to slang

  • set theory — noun The mathematical theory of sets …   Wiktionary

  • set theory — noun Date: 1936 a branch of mathematics or of symbolic logic that deals with the nature and relations of sets • set theoretic adjective …   New Collegiate Dictionary

  • set theory — /ˈsɛt θɪəri/ (say set thearree) noun the branch of mathematics that deals with the properties of sets (def. 78) …   Australian English dictionary

  • set theory — branch of mathematics that deals with the grouping of theorems and proving them …   English contemporary dictionary

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