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In set theory, axiomatic set theory is a rigorous reformulation of set theory, which in its traditional form has now become described as naive set theory. more...

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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.

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rigor in proofs. At the same time the basic concepts of set theory are used throughout mathematics, the subject is pursued in its own right as a specialty by a comparatively small group of mathematicians and logicians. It should be mentioned that there are also mathematicians using and promoting different approaches to the foundations of mathematics.

The basic concepts of set theory are set and membership. A set is thought of as any collection of objects, called the members (or elements) of the set. In mathematics, the members of sets are any mathematical objects, and in particular can themselves be sets. Thus one speaks of the set N of natural numbers { 0, 1, 2, 3, 4, ... }, the set 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 } which has as members the numbers 0 and 2 and the set N.

Initially, what is now known as "naive" or "intuitive" set theory was developed. As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox. To address these problems, set theory had to be re-constructed, using an axiomatic approach.

The origins of rigorous set theory

The important idea of Cantor's, which got set theory going as a new field of study, was to 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. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers does not have the same cardinality as N or Q, but a larger one (it is said to be uncountable). Cantor gave two proofs that R is not countable, and the second of these, using what is known as the diagonal construction, has been extraordinarily influential and has had many applications in logic and mathematics.

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