Uncountable set

In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers.

Examples of uncountable sets include the set ⁠

R

{\displaystyle \mathbb {R} }

⁠ of all real numbers and the set of all subsets of the natural numbers.

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

ℵ

0

{\displaystyle \aleph _{0}}

(aleph-null).

The set X has cardinality strictly greater than

ℵ

0

{\displaystyle \aleph _{0}}

.

The first three of these characterizations can be proved 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.

Properties

If an uncountable set X is a subset of set Y, then Y is uncountable.