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# 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.
```