SetTheory

Set theory is a part of mathematics created by the German mathematician Georg Cantor at the end of the 19th 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. At the same time the basic concepts of set theory are used throughout mathematics, while the subject is pursued in its own right as a speciality 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 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 of natural numbers (0,1,2,..), the set of real numbers, the set of functions from the natural numbers to the natural numbers, but also e.g. of the set {0,2,N} which has as members the numbers 0 and 2 and the set N of natural numbers. Cantor's basic discovery, which got set theory going as a new field of study, 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, 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), but the set R of real numbers does not have the same cardinality as N or Q. 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 manifold applications in logic and mathematics.

The appearance around the turn of the century of the so-called set-theoretical paradoxes prompted the formulation in 1908 by Zermelo of an axiomatic theory of sets, and the theory of sets now most often studied and used is the theory called Zermelo-Fraenkel or ZF.

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Does the above have the same content as the /OldVersion? Please check.

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No. The old page was math with some disagreement. This is meta-math, so to speak. Please ask JoshuaGrosse I believe it was his page.

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Well, then, something's got to be done. Actually, I like the above article. It's the sort of thing that's needed as an introductory article about set theory.