Sets:Uncountable - Revision history

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In [[mathematics]], an '''uncountable set''' (or '''uncountably infinite set''')<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> is an [[infinite set]] that contains too many [[Element (mathematics)|elements]] to be [[countable set|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 number]]s.

==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 exist 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 <math>\aleph_0</math> ([[aleph number|aleph-null]], the cardinality of the [[natural number]]s).
* The set ''X'' has cardinality strictly greater than <math>\aleph_0</math>.

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.

==Properties==
* If an uncountable set ''X'' is a subset of set ''Y'', then ''Y'' is uncountable.

== Examples ==

The best known example of an uncountable set is the set '''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 and the set of all [[subset]]s of the set of natural numbers. The cardinality of '''R''' is often called the [[cardinality of the continuum]], and denoted by <math>\mathfrak{c} </math>,<ref>{{Cite web|date=2020-04-11|title=Comprehensive List of Set Theory Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/set-theory-symbols/|access-date=2020-09-05|website=Math Vault|language=en-US}}</ref> 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 '''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 [[Function (mathematics)|function]]s from '''R''' to '''R'''. This set is even "more uncountable" than '''R''' in the sense that the cardinality of this set is <math>\beth_2</math> ([[beth two|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" /> 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]]).

==Without the axiom of choice==
{{main|Dedekind-infinite set}}
Without the [[axiom of choice]], there might exist cardinalities [[Comparability|incomparable]] to <math>\aleph_0</math> (namely, the cardinalities of [[Dedekind-finite]] infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.

If the axiom of choice holds, the following conditions on a cardinal <math>\kappa</math> are equivalent:
*<math>\kappa \nleq \aleph_0;</math>
*<math>\kappa > \aleph_0;</math> and
*<math>\kappa \geq \aleph_1</math>, where <math>\aleph_1 = |\omega_1 |</math> and <math>\omega_1</math> is the least [[initial ordinal]] greater than <math>\omega.</math>

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.

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 However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Uncountable_set Uncountable set], Wikipedia
+Content obtained and/or adapted from:

← Older revision		Revision as of 19:24, 13 October 2021
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	However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.		However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Uncountable_set Uncountable set], Wikipedia

@@ Line 1: / Line 1: @@
-In [[mathematics]], an '''uncountable set''' (or '''uncountably infinite set''')<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> is an [[infinite set]] that contains too many [[Element (mathematics)|elements]] to be [[countable set|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 number]]s.
+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.
 ==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.
+* 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 exist 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''.
+* ''X'' is nonempty and for every ω-sequence of elements of ''X'', there exist 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 <math>\aleph_0</math> ([[aleph number|aleph-null]], the cardinality of the [[natural number]]s).
+* The cardinality of ''X'' is neither finite nor equal to <math>\aleph_0</math> (aleph-null, the cardinality of the natural numbers).
 * The set ''X'' has cardinality strictly greater than <math>\aleph_0</math>.
-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 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.
 ==Properties==
@@ Line 16: / Line 16: @@
 == Examples ==
-The best known example of an uncountable set is the set '''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  and the set of all [[subset]]s of the set of natural numbers. The cardinality of '''R''' is often called the [[cardinality of the continuum]], and denoted by <math>\mathfrak{c} </math>,<ref>{{Cite web|date=2020-04-11|title=Comprehensive List of Set Theory Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/set-theory-symbols/|access-date=2020-09-05|website=Math Vault|language=en-US}}</ref> or <math>2^{\aleph_0}</math>, or <math>\beth_1</math> ([[cardinality of the continuum|beth-one]]).
+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 <math>\mathfrak{c} </math>, or <math>2^{\aleph_0}</math>, or <math>\beth_1</math> (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.
+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 [[Function (mathematics)|function]]s from '''R''' to '''R'''. This set is even "more uncountable" than '''R''' in the sense that the cardinality of this set is <math>\beth_2</math> ([[beth two|beth-two]]), which is larger than <math>\beth_1</math>.
+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 <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" /> 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]]).
+A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω<sub>1</sub>. The cardinality of Ω is denoted <math>\aleph_1</math> (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 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).
 ==Without the axiom of choice==
-{{main|Dedekind-infinite set}}
-Without the [[axiom of choice]], there might exist cardinalities [[Comparability|incomparable]] to <math>\aleph_0</math> (namely, the cardinalities of [[Dedekind-finite]] infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.
+Without the axiom of choice, there might exist cardinalities incomparable to <math>\aleph_0</math> (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.
 If the axiom of choice holds, the following conditions on a cardinal <math>\kappa</math> are equivalent:
 *<math>\kappa \nleq \aleph_0;</math>
 *<math>\kappa > \aleph_0;</math> and
-*<math>\kappa \geq \aleph_1</math>, where <math>\aleph_1 = |\omega_1 |</math> and <math>\omega_1</math> is the least [[initial ordinal]] greater than <math>\omega.</math>
+*<math>\kappa \geq \aleph_1</math>, where <math>\aleph_1 = |\omega_1 |</math> and <math>\omega_1</math> is the least initial ordinal greater than <math>\omega.</math>
 However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.