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{{Short description|Set that is not a finite set}}
[[File:Real numbers.svg|alt=Set Theory Image|thumb|Set Theory Image]]
▲{{Refimprove|date=September 2011}}
In [[set theory]], an '''infinite set''' is a [[Set (mathematics)|set]] that is not a [[finite set]]. [[Infinity|Infinite]] sets may be [[countable set|countable]] or [[uncountable set|uncountable]].<ref name=Bagaria/>
==Properties==
The set of [[natural numbers]] (whose existence is postulated by the [[axiom of infinity]]) is infinite.<ref name=
A set is infinite if and only if for every natural number, the set has a [[subset]] whose [[cardinality]] is that natural number.<ref>{{
If the [[axiom of choice]] holds, then a set is infinite if and only if it includes a countable infinite subset.
If a [[set of sets]] is infinite or contains an infinite element, then its union is infinite. The [[power set]] of an infinite set is infinite.<ref name=":1" /> Any [[subset|superset]] of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped ''[[onto]]'' an infinite set is infinite. The [[Cartesian product]] of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets, each containing at least two elements, is either empty or infinite; if the axiom of choice holds, then it is infinite.
If an infinite set is a [[well-ordered set]], then it must have a nonempty, nontrivial subset that has no greatest element.
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If an infinite set is a [[well-orderable set]], then it has many well-orderings which are non-isomorphic.
==History==
Important ideas discussed by David Burton in his book ''The History of Mathematics: An Introduction'' include how to define "elements" or parts of a set, how to define unique elements in the set, and how to prove infinity.<ref name=":2">{{Cite book |last=Burton |first=David |title=The History of Mathematics: An Introduction |publisher=McGraw Hill |year=2007 |isbn=9780073051895 |edition=6th |___location=Boston |pages=666–689 |language=en}}</ref> Burton also discusses proofs for different types of infinity, including countable and uncountable sets.<ref name=":2" /> Topics used when comparing infinite and finite sets include [[ordered set]]s, cardinality, equivalency, [[coordinate plane]]s, [[universal set]]s, mapping, subsets, continuity, and [[Transcendental number theory|transcendence]].<ref name=":2" /> [[Georg Cantor|Cantor's]] set ideas were influenced by trigonometry and irrational numbers. Other key ideas in infinite set theory mentioned by Burton, Paula, Narli and Rodger include real numbers such as [[pi|{{pi}}]], integers, and [[Euler's number]].<ref name=":2" /><ref>{{Cite journal |last1=Pala |first1=Ozan |last2=Narli |first2=Serkan |date=2020-12-15 |title=Role of the Formal Knowledge in the Formation of the Proof Image: A Case Study in the Context of Infinite Sets |journal=Turkish Journal of Computer and Mathematics Education |language=en |volume=11 |issue=3 |pages=584–618 |doi=10.16949/turkbilmat.702540|s2cid=225253469 |doi-access=free }}</ref><ref name=":3">{{Cite book |last=Rodgers |first=Nancy |title=Learning to reason: an introduction to logic, sets and relations |date=2000 |publisher=Wiley |isbn=978-1-118-16570-6 |___location=New York |oclc=757394919}}</ref>
Both Burton and Rogers use finite sets to start to explain infinite sets using proof concepts such as mapping, proof by induction, or proof by contradiction.<ref name=":2" /><ref name=":3" /> [[Tree (set theory)|Mathematical trees]] can also be used to understand infinite sets.<ref>{{Cite journal |last1=Gollin |first1=J. Pascal |last2=Kneip |first2=Jakob |date=2021-04-01 |title=Representations of Infinite Tree Sets |journal=Order |language=en |volume=38 |issue=1 |pages=79–96 |doi=10.1007/s11083-020-09529-0 |s2cid=201646182 |issn=1572-9273|doi-access=free |arxiv=1908.10327 }}</ref> Burton also discusses proofs of infinite sets including ideas such as unions and subsets.<ref name=":2" />
In Chapter 12 of ''The History of Mathematics: An Introduction'', Burton emphasizes how mathematicians such as [[Ernst Zermelo|Zermelo]], [[Dedekind]], [[Galileo]], [[Leopold Kronecker|Kronecker]], Cantor, and [[Bernard Bolzano|Bolzano]] investigated and influenced infinite set theory. Many of these mathematicians either debated infinity or otherwise added to the ideas of infinite sets. Potential historical influences, such as how Prussia's history in the 1800s, resulted in an increase in scholarly mathematical knowledge, including Cantor's theory of infinite sets.<ref name=":2" />
One potential application of infinite set theory is in genetics and biology.<ref>{{Cite journal |last1=Shelah |first1=Saharon |last2=Strüngmann |first2=Lutz |date=2021-06-01 |title=Infinite combinatorics in mathematical biology |journal=Biosystems |language=en |volume=204 |pages=104392 |doi=10.1016/j.biosystems.2021.104392 |pmid=33731280 |s2cid=232298447 |issn=0303-2647|doi-access=free |bibcode=2021BiSys.20404392S }}</ref>
==Examples==
===Countably infinite sets===
The set of all [[integer]]s, {...,
The set of all [[
===Uncountably infinite sets===
The set of all [[real number]]s is an uncountably infinite set. The set of all [[
The set of all subsets of the integers is uncountably infinite.
==See also==
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