Editing Uncountable set (section)

==Characterizations==

There are many equivalent characterizations of uncountability.  A set ''X'' is uncountable if and only if any of the following conditions hold:
* There is no [[injective function]] (hence no [[bijection]]) from ''X'' to the set of natural numbers.
* ''X'' is nonempty and for every ω-[[sequence]] of elements of ''X'', there exists at least one element of X not included in it. That is, ''X'' is nonempty and there is no [[surjective function]] from the natural numbers to ''X''.
* The [[cardinality]] of ''X'' is neither finite nor equal to <math>\aleph_0</math> ([[aleph number|aleph-null]]).
* The set ''X'' has cardinality strictly greater than <math>\aleph_0</math>.

The first three of these characterizations can be proven equivalent in [[Zermelo–Fraenkel set theory]] without the [[axiom of choice]], but the equivalence of the third and fourth cannot be proved without additional choice principles.