Editing Set (mathematics) (section)

===Infinite cardinalities===
The cardinality of an infinite set is commonly represented by a [[cardinal number]], exactly as the number of elements of a finite set is represented by a natural numbers. The definition of cardinal numbers is too technical for this article; however, many properties of cardinalities can be dealt without referring to cardinal numbers, as follows.

Two sets {{tmath|S}} and {{tmath|T}} have the same cardinality if there exists a one-to-one correspondence ([[bijection]]) between them. This is denoted <math>|S|=|T|,</math> and would be an [[equivalence relation]] on sets, if a set of all sets would exist.

For example, the natural numbers and the even natural numbers have the same cardinality, since multiplication by two provides such a bijection. Similarly, the [[interval (mathematics)|interval]] {{tmath|(-1, 1)}} and the set of all real numbers have the same cardinality, a bijection being provided by the function {{tmath|x\mapsto \tan(\pi x/2)}}.

Having the same cardinality of a [[proper subset]] is a characteristic property of infinite sets: ''a set is infinite if and only if it has the same cardinality as one of its proper subsets.''
So, by the above example, the natural numbers form an infinite set.<ref name="Lucas1990"/>

Besides equality, there is a natural inequality between cardinalities: a set {{tmath|S}} has a cardinality smaller than or equal to the cardinality of another set {{tmath|T}} if there is an [[injection (mathematics)|injection]] frome {{tmath|S}} to {{tmath|T}}. This is denoted <math>|S|\le |T|.</math>

[[Schröder–Bernstein theorem]] implies that <math>|S|\le |T|</math> and <math>|T|\le |S|</math> imply <math>|S|= |T|.</math> Also, one has <math>|S|\le |T|,</math> if and only if there is a surjection from {{tmath|T}} to {{tmath|S}}. For every two sets {{tmath|S}} and {{tmath|T}}, one has either <math>|S|\le |T|</math> or <math>|T|\le |S|.</math>{{efn|This property is equivalent to the [[axiom of choice]].}} So, inequality of cardinalities is a [[total order]].

The cardinality of the set {{tmath|\N}} of the natural numbers, denoted <math>|\N|=\aleph_0,</math> is the smallest infinite cardinality. This means that if {{tmath|S}} is a set of natural numbers, then either {{tmath|S}} is finite or <math>|S|=|\N|.</math>

Sets with cardinality less than or equal to <math>|\N|=\aleph_0</math> are called ''[[countable set]]s''; these are either finite sets or ''[[countably infinite set]]s'' (sets of cardinality <math>\aleph_0</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than <math>\aleph_0</math> are called ''[[uncountable set]]s''.

[[Cantor's diagonal argument]] shows that, for every set {{tmath|S}}, its power set (the set of its subsets) {{tmath|2^S}} has a greater cardinality:
<math display=block>|S|<\left|2^S \right|.</math>
This implies that there is no greatest cardinality.