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 ⁠${\displaystyle \mathbb {R} }$⁠ of all real numbers and set of all subsets of the natural numbers.

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

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.

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

The best known example of an uncountable set is the set ⁠${\displaystyle \mathbb {R} }$⁠ 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 ⁠${\displaystyle \mathbb {N} }$⁠ (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set of natural numbers. The cardinality of ⁠${\displaystyle \mathbb {R} }$⁠ is often called the cardinality of the continuum, and denoted by ${\displaystyle {\mathfrak {c}}}$, or ${\displaystyle 2^{\aleph _{0}}}$, or ${\displaystyle \beth _{1}}$ (beth-one).

The Cantor set is an uncountable subset of ⁠${\displaystyle \mathbb {R} }$⁠. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one (⁠${\displaystyle \mathbb {R} }$⁠ has dimension one). This is an example of the following fact: any subset of ⁠${\displaystyle \mathbb {R} }$⁠ of Hausdorff dimension strictly greater than zero must be uncountable.

Another example of an uncountable set is the set of all functions from ⁠${\displaystyle \mathbb {R} }$⁠ to ⁠${\displaystyle \mathbb {R} }$⁠. This set is even "more uncountable" than ⁠${\displaystyle \mathbb {R} }$⁠ in the sense that the cardinality of this set is ${\displaystyle \beth _{2}}$ (beth two), which is larger than ${\displaystyle \beth _{1}}$.