Uncountable Set - Examples

Examples

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, or (beth-one).

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

Another example of an uncountable set is the set of all functions from R to R. This set is even "more uncountable" than R in the sense that the cardinality of this set is (beth-two), which is larger than .

A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω₁. The cardinality of Ω is denoted (aleph-one). It can be shown, using the axiom of choice, that is the smallest uncountable cardinal number. Thus either, the cardinality of the reals, is equal to or it is strictly larger. Georg Cantor was the first to propose the question of whether is equal to . In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that is now called the continuum hypothesis and is known to be independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice).