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set theory
A branch of mathematics created by Georg Cantor at the end of the nineteenth century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rigor in proofs. Cantor's basic discovery was that if we define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. The appearance around the turn of the century of the so-called set-theoretical paradoxes, such as Russell's paradox, prompted the formulation in 1908 by Ernst Zermelo of an axiomatic theory of sets. The axioms for set theory now most often studied and used are those called the Zermelo-Fraenkel axioms, usually together with the axiom of choice. The Zermelo-Fraenkel axioms are commonly abbreviated to ZF, or ZFC if the axiom of choice is included. An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, such as numbers, must be subsequently defined in terms of sets.
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