Editing Uncountable set (section)

== Examples ==

The best known example of an uncountable set is the set {{tmath|\R}} of all [[real number]]s; [[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 [[sequence]]s of [[natural number]]s {{tmath|\N}} (see: {{OEIS|A102288}}), and the [[Powerset|set of all subsets]] of the set of natural numbers. The cardinality of {{tmath|\R}} is often called the [[cardinality of the continuum]], and denoted by <math>\mathfrak{c} </math>, or <math>2^{\aleph_0}</math>, or <math>\beth_1</math> ([[cardinality of the continuum|beth-one]]).

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

Another example of an uncountable set is the set of all [[Function (mathematics)|function]]s from {{tmath|\R}} to {{tmath|\R}}. This set is even "more uncountable" than {{tmath|\R}} in the sense that the cardinality of this set is <math>\beth_2</math> ([[beth two]]), which is larger than <math>\beth_1</math>.

A more abstract example of an uncountable set is the set of all countable [[ordinal number]]s, denoted by Ω or ω<sub>1</sub>.<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Uncountably Infinite|url=https://mathworld.wolfram.com/UncountablyInfinite.html|access-date=2020-09-05|website=mathworld.wolfram.com|language=en}}</ref> The cardinality of Ω is denoted <math>\aleph_1</math> ([[aleph number|aleph-one]]). It can be shown, using the [[axiom of choice]], that <math>\aleph_1</math> is the ''smallest'' uncountable cardinal number. Thus either <math>\beth_1</math>, the cardinality of the reals, is equal to <math>\aleph_1</math> or it is strictly larger. [[Georg Cantor]] was the first to propose the question of whether <math>\beth_1</math> is equal to <math>\aleph_1</math>.  In 1900, [[David Hilbert]] posed this question as the first of his [[Hilbert's problems|23 problems]]. The statement that <math>\aleph_1 = \beth_1</math> 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]]).