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In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers.
A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number.
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. Any 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.
In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets.
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. Burton also discusses proofs for different types of infinity, including countable and uncountable sets. Topics used when comparing infinite and finite sets include ordered sets, cardinality, equivalency, coordinate planes, universal sets, mapping, subsets, continuity, and transcendence.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 π, integers, and Euler's number.
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.Mathematical trees can also be used to understand infinite sets. Burton also discusses proofs of infinite sets including ideas such as unions and subsets.
In Chapter 12 of The History of Mathematics: An Introduction, Burton emphasizes how mathematicians such as Zermelo, Dedekind, Galileo, Kronecker, Cantor, and 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.
One potential application of infinite set theory is in genetics and biology.
Examples
Countably infinite sets
The set of all integers, {..., −1, 0, 1, 2, ...} is a countably infinite set. The set of all even integers is also a countably infinite set, even if it is a proper subset of the integers.
The set of all rational numbers is a countably infinite set as there is a bijection to the set of integers.
Uncountably infinite sets
The set of all real numbers is an uncountably infinite set. The set of all irrational numbers is also an uncountably infinite set.
The set of all subsets of the integers is uncountably infinite.
See also
References
- Bagaria, Joan (2019), "Set Theory", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-11-30
- Boolos, George (1998). Logic, Logic, and Logic (illustrated ed.). Harvard University Press. p. 262. ISBN 978-0-674-53766-8.
- Caldwell, Chris. "The Prime Glossary — Infinite". primes.utm.edu. Retrieved 2019-11-29.
- Boolos, George (1994), "The advantages of honest toil over theft", Mathematics and mind (Amherst, MA, 1991), Logic Comput. Philos., Oxford Univ. Press, New York, pp. 27–44, MR 1373892. See in particular pp. 32–33.
- Burton, David (2007). The History of Mathematics: An Introduction (6th ed.). Boston: McGraw Hill. pp. 666–689. ISBN 9780073051895.
- Pala, Ozan; Narli, Serkan (2020-12-15). "Role of the Formal Knowledge in the Formation of the Proof Image: A Case Study in the Context of Infinite Sets". Turkish Journal of Computer and Mathematics Education (TURCOMAT). 11 (3): 584–618. doi:10.16949/turkbilmat.702540. S2CID 225253469.
- Rodgers, Nancy (2000). Learning to reason: an introduction to logic, sets and relations. New York: Wiley. ISBN 978-1-118-16570-6. OCLC 757394919.
- Gollin, J. Pascal; Kneip, Jakob (2021-04-01). "Representations of Infinite Tree Sets". Order. 38 (1): 79–96. arXiv:1908.10327. doi:10.1007/s11083-020-09529-0. ISSN 1572-9273. S2CID 201646182.
- Shelah, Saharon; Strüngmann, Lutz (2021-06-01). "Infinite combinatorics in mathematical biology". Biosystems. 204: 104392. Bibcode:2021BiSys.20404392S. doi:10.1016/j.biosystems.2021.104392. ISSN 0303-2647. PMID 33731280. S2CID 232298447.
External links
- A Crash Course in the Mathematics Of Infinite Sets
This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Infinite set news newspapers books scholar JSTOR September 2011 Learn how and when to remove this message In set theory an infinite set is a set that is not a finite set Infinite sets may be countable or uncountable Set Theory ImagePropertiesThe set of natural numbers whose existence is postulated by the axiom of infinity is infinite It is the only set that is directly required by the axioms to be infinite The existence of any other infinite set can be proved in Zermelo Fraenkel set theory ZFC but only by showing that it follows from the existence of the natural numbers A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number 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 Any 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 In ZF a set is infinite if and only if the power set of its power set is a Dedekind infinite set having a proper subset equinumerous to itself If the axiom of choice is also true then infinite sets are precisely the Dedekind infinite sets If an infinite set is a well orderable set then it has many well orderings which are non isomorphic HistoryImportant 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 Burton also discusses proofs for different types of infinity including countable and uncountable sets Topics used when comparing infinite and finite sets include ordered sets cardinality equivalency coordinate planes universal sets mapping subsets continuity and transcendence 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 p integers and Euler s number 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 Mathematical trees can also be used to understand infinite sets Burton also discusses proofs of infinite sets including ideas such as unions and subsets In Chapter 12 of The History of Mathematics An Introduction Burton emphasizes how mathematicians such as Zermelo Dedekind Galileo Kronecker Cantor and 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 One potential application of infinite set theory is in genetics and biology ExamplesCountably infinite sets The set of all integers 1 0 1 2 is a countably infinite set The set of all even integers is also a countably infinite set even if it is a proper subset of the integers The set of all rational numbers is a countably infinite set as there is a bijection to the set of integers Uncountably infinite sets The set of all real numbers is an uncountably infinite set The set of all irrational numbers is also an uncountably infinite set The set of all subsets of the integers is uncountably infinite See alsoAleph number Cardinal number Ordinal numberReferencesBagaria Joan 2019 Set Theory in Zalta Edward N ed The Stanford Encyclopedia of Philosophy Fall 2019 ed Metaphysics Research Lab Stanford University retrieved 2019 11 30 Boolos George 1998 Logic Logic and Logic illustrated ed Harvard University Press p 262 ISBN 978 0 674 53766 8 Caldwell Chris The Prime Glossary Infinite primes utm edu Retrieved 2019 11 29 Boolos George 1994 The advantages of honest toil over theft Mathematics and mind Amherst MA 1991 Logic Comput Philos Oxford Univ Press New York pp 27 44 MR 1373892 See in particular pp 32 33 Burton David 2007 The History of Mathematics An Introduction 6th ed Boston McGraw Hill pp 666 689 ISBN 9780073051895 Pala Ozan Narli Serkan 2020 12 15 Role of the Formal Knowledge in the Formation of the Proof Image A Case Study in the Context of Infinite Sets Turkish Journal of Computer and Mathematics Education TURCOMAT 11 3 584 618 doi 10 16949 turkbilmat 702540 S2CID 225253469 Rodgers Nancy 2000 Learning to reason an introduction to logic sets and relations New York Wiley ISBN 978 1 118 16570 6 OCLC 757394919 Gollin J Pascal Kneip Jakob 2021 04 01 Representations of Infinite Tree Sets Order 38 1 79 96 arXiv 1908 10327 doi 10 1007 s11083 020 09529 0 ISSN 1572 9273 S2CID 201646182 Shelah Saharon Strungmann Lutz 2021 06 01 Infinite combinatorics in mathematical biology Biosystems 204 104392 Bibcode 2021BiSys 20404392S doi 10 1016 j biosystems 2021 104392 ISSN 0303 2647 PMID 33731280 S2CID 232298447 External linksA Crash Course in the Mathematics Of Infinite Sets
