In mathematics, axiomatic set theory is a reformulation of set theory as a rigorous axiomatic first order theory. The intention was to create a set theory free of the antinomies (paradoxes) that afflicted naive set theory. Axiomatic set theory has since become the canonical approach to the foundation of mathematics, and an active research area in its own right, albeit by relatively few mathematicians.
The undefined concepts of set theory are "set" and "set membership." A set is any collection of objects, called the members (or elements) of the set. An object can be anything physical or abstract of interest to humans; in particular, an object can be (and often) is itself a set. In mathematics, the members of sets are mathematical constructions. Thus one speaks of the set N of natural numbers { 0, 1, 2, 3, 4, ... }, the set R of real numbers, and the set of functions from the natural numbers to the natural numbers; but also, for example, of the set { 0, 2, N}, whose members are 0, 2 and the set N.
Initially controversial, set theory has since become the main foundational theory of modern mathematics. (There is active research in foundational theories other than set theory.) The mathematical objects (numbers, functions, etc.,) of algebra, analysis, topology, etc., are now normally defined as particular sets having particular properties. It is commonly thought (but not empirically demonstrated or logically demonstrable) that most or all of contemporary mathematics consists of theorems that can be proved using axiomatic set theory.
Reference:
http://en.wikipedia.org/wiki/Axiomatic_set_theory
