Editing Countable set (section)

==Definition==

A set <math>S</math> is ''countable'' if:
* Its [[cardinality]] <math>|S|</math> is less than or equal to <math>\aleph_0</math> ([[aleph-null]]), the cardinality of the set of [[natural numbers]] <math>\N</math>.<ref name=Yaqub/>
* There exists an [[injective function]] from <math>S</math> to <math>\N</math>.<ref name=Singh>{{cite book |last1=Singh |first1=Tej Bahadur |title=Introduction to Topology |date=17 May 2019 |publisher=Springer |isbn=978-981-13-6954-4 |page=422 |url=https://books.google.com/books?id=UQiZDwAAQBAJ&pg=PA422 |language=en}}</ref><ref name=Katzourakis>{{cite book |last1=Katzourakis |first1=Nikolaos |last2=Varvaruca |first2=Eugen |title=An Illustrative Introduction to Modern Analysis |date=2 January 2018 |publisher=CRC Press |isbn=978-1-351-76532-9 |url=https://books.google.com/books?id=jBFFDwAAQBAJ&pg=PT15 |language=en}}</ref>
* <math>S</math> is empty or there exists a [[surjective function]] from <math>\N</math> to <math>S</math>.<ref name=Katzourakis/>
* There exists a [[bijective]] mapping between <math>S</math> and a subset of <math>\N</math>.<ref>{{harvnb|Halmos|1960|loc=p. 91}}</ref>
* <math>S</math> is either [[Finite set|finite]] (<math>|S|<\aleph_0</math>) or countably infinite.<ref name="Lang"/>
All of these definitions are equivalent.

A set <math>S</math> is ''countably [[infinite set|infinite]]'' if:
* Its cardinality <math>|S|</math> is exactly <math>\aleph_0</math>.<ref name=Yaqub/>
* There is an injective and surjective (and therefore [[bijection|bijective]]) mapping between <math>S</math> and <math>\N</math>.
* <math>S</math> has a [[One-one correspondence|one-to-one correspondence]] with <math>\N</math>.<ref>{{harvnb|Kamke|1950|loc=p. 2}}</ref>
* The elements of <math>S</math> can be arranged in an infinite sequence <math>a_0, a_1, a_2, \ldots</math>, where <math>a_i</math> is distinct from <math>a_j</math> for <math>i\neq j</math> and every element of <math>S</math> is listed.<ref>{{cite book |last1=Dlab |first1=Vlastimil |last2=Williams |first2=Kenneth S. |title=Invitation To Algebra: A Resource Compendium For Teachers, Advanced Undergraduate Students And Graduate Students In Mathematics |date=9 June 2020 |publisher=World Scientific |isbn=978-981-12-1999-3 |page=8 |url=https://books.google.com/books?id=l9rrDwAAQBAJ&pg=PA8 |language=en}}</ref><ref>{{harvnb|Tao|2016|p=182}}</ref>

A set is ''[[uncountable]]'' if it is not countable, i.e. its cardinality is greater than <math>\aleph_0</math>.<ref name=Yaqub>{{cite book |last1=Yaqub |first1=Aladdin M. |title=An Introduction to Metalogic |date=24 October 2014 |publisher=Broadview Press |isbn=978-1-4604-0244-3 |url=https://books.google.com/books?id=cyljCAAAQBAJ&pg=PT187 |language=en}}</ref>