Uncountably infinite

In mathematics, an uncountable set (or uncountably infinite set) 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 that of the set of all natural numbers.

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds:

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.

The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or ${\displaystyle 2^{\aleph _{0}}}$, or ${\displaystyle \beth _{1}}$ (beth-one).

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