Editing Cardinality (section)

{{Short description|Size of a set}}
{{Other uses}}
[[File:Apples and Oranges v2.png|thumb|318x318px|A bijection, comparing a set of apples to a set of oranges, showing they have the same cardinality.]]
The '''cardinality''' of a [[finite set]] is the number of its elements, and is therefore a measure of size of the set. Since the discovery by [[Georg Cantor]], in the late 19th century, of different sizes of [[infinite set]]s, the term ''cardinality'' was coined for generalizing to [[infinite sets]] the concept of the number of elements.

More precisely, two sets have the same cardinality if there exists a [[one-to-one correspondence]] between them. In the case of finite sets, the common operation of ''counting'' consists of establishing a one-to-one correspondence between a given set and the set of the {{tmath|n}} first [[natural number]]s, for some natural number {{tmath|n}}. In this case, {{tmath|n}} is the cardinality of the set.

A set is ''infinite'' if the counting operation never finishes, that is, if its cardinality is not a natural number. The basic example of an infinite set is the set of all natural numbers.

The great discovery of Cantor is that infinite sets of apparently different size may have the same cardinality, but that there are infinitely many possible cardinalities. For example, the [[even number]]s, the [[prime number]]s and the [[polynomial]]s whose coefficients are [[rational number]] have the same cardinality as the natural numbers. The set of the [[real number]]s has a greater cardinality than the natural numbers, and has the same cardinality as the interval {{tmath|[0,1]}} and as every [[Euclidean space]] of any dimension. For every set, its [[power set]] (the set of all its subsets) has a greater cardinality.

Cardinalities are represented with [[cardinal number]]s, which are specific sets of a given cardinality, which have been chosen once for all. Some infinite cardinalities have been given a specific name, such as {{tmath|\aleph_0}} for the cardinality of the natural numbers and {{tmath|\mathfrak c}}, the [[cardinality of the continuum]], for the cardinality of the real numbers.