Infinite set

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.^[1]

Properties

The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite.^[1] 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.^[2]

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.^[3] 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.^[4] 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.^[5] Burton also discusses proofs for different types of infinity, including countable and uncountable sets.^[5] Topics used when comparing infinite and finite sets include ordered sets, cardinality, equivalency, coordinate planes, universal sets, mapping, subsets, continuity, and transcendence.^[5] 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.^[5]^[6]^[7]

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.^[5]^[7] Mathematical trees can also be used to understand infinite sets.^[8] Burton also discusses proofs of infinite sets including ideas such as unions and subsets.^[5]

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.^[5]

One potential application of infinite set theory is in genetics and biology.^[9]

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.^[3]

The set of all rational numbers is a countably infinite set as there is a bijection to the set of integers.^[3]

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.^[3]

References

^ ^a ^b 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.
^ ^a ^b ^c ^d 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.
^ ^a ^b ^c ^d ^e ^f ^g 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.
^ ^a ^b 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. 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

[Bagaria-1] 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

[2] Boolos, George (1998). Logic, Logic, and Logic (illustrated ed.). Harvard University Press. p. 262. ISBN 978-0-674-53766-8.

[:1-3] Caldwell, Chris. "The Prime Glossary — Infinite". primes.utm.edu. Retrieved 2019-11-29.

[4] 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.

[:2-5] ^ ^a ^b ^c ^d ^e ^f ^g Burton, David (2007). The History of Mathematics: An Introduction (6th ed.). Boston: McGraw Hill. pp. 666–689. ISBN 9780073051895.

[6] 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.

[:3-7] Rodgers, Nancy (2000). Learning to reason: an introduction to logic, sets and relations. New York: Wiley. ISBN 978-1-118-16570-6. OCLC 757394919.

[8] 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.

[9] Shelah, Saharon; Strüngmann, Lutz (2021-06-01). "Infinite combinatorics in mathematical biology". Biosystems. 204: 104392. doi:10.1016/j.biosystems.2021.104392. ISSN 0303-2647. PMID 33731280. S2CID 232298447.

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v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo