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 sets of real numbers, the sets 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.