Editing Cardinal number (section)

{{Short description|Size of a possibly infinite set}}
{{about|the mathematical concept|number words indicating quantity ("three" apples, "four" birds, etc.)|Cardinal numeral}}

[[File:Bijection.svg|thumb|200px|A [[bijective function]], ''f'': ''X'' → ''Y'', from set ''X'' to set ''Y '' demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.]]

[[File:Aleph0.svg|thumb|right|150px|[[Aleph-null]], the smallest infinite cardinal]]

In [[mathematics]], a '''cardinal number''', or '''cardinal''' for short, is what is commonly called the number of elements of a [[set (mathematics)|set]]. In the case of a [[finite set]], its cardinal number, or [[cardinality]] is therefore a [[natural number]]. For dealing with the case of [[infinite set]]s, the [[transfinite number|infinite cardinal number]]s have been introduced, which are often denoted with the [[Hebrew alphabet|Hebrew letter]] <math>\aleph</math> ([[Aleph (Hebrew)|aleph]]) marked with subscript indicating their rank among the infinite cardinals.

Cardinality is defined in terms of [[bijective function]]s. Two sets have the same cardinality [[if, and only if]], there is a [[one-to-one correspondence]] (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to [[Georg Cantor]] shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of [[real number]]s is greater than the cardinality of the set of natural numbers. It is also possible for a [[proper subset]] of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.

There is a [[transfinite sequence]] of cardinal numbers:
:<math>0, 1, 2, 3, \ldots, n, \ldots ; \aleph_0, \aleph_1, \aleph_2, \ldots, \aleph_{\alpha}, \ldots.\ </math>
This sequence starts with the [[natural number]]s including zero (finite cardinals), which are followed by the [[aleph number]]s. The aleph numbers are indexed by [[ordinal number]]s. If the [[axiom of choice]] is true, this transfinite sequence includes every cardinal number. If the axiom of choice is not true (see {{slink|Axiom of choice#Independence}}), there are infinite cardinals that are not aleph numbers.

[[Cardinality]] is studied for its own sake as part of [[set theory]]. It is also a tool used in branches of mathematics including [[model theory]], [[combinatorics]], [[abstract algebra]] and [[mathematical analysis]]. In [[category theory]], the cardinal numbers form a [[Skeleton (category theory)|skeleton]] of the [[category of sets]].