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If an infinite set is a [[well-orderable set]], then it has many well-orderings which are non-isomorphic.
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 sets, cardinality, equivalency, coordinate planes, universal sets, mapping, subsets, continuity, and 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 |url=https://dergipark.org.tr/tr/pub/turkbilmat/issue/58294/702540 |journal=Turkish Journal of Computer and Mathematics Education (TURCOMAT) |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" /> 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 }}</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 1800's, 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 |url=https://www.sciencedirect.com/science/article/pii/S0303264721000496 |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 }}</ref>
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