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Inset theory, aninfinite set is aset that is not afinite set.Infinite sets may becountable oruncountable.[1]
The set ofnatural numbers (whose existence is postulated by theaxiom of infinity) is infinite.[1] It is the only set that is directly required by theaxioms to be infinite. The existence of any other infinite set can be proved inZermelo–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 asubset whosecardinality is that natural number.[2]
If theaxiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset.
If aset of sets is infinite or contains an infinite element, then its union is infinite. Thepower set of an infinite set is infinite.[3] Anysuperset 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 mappedonto an infinite set is infinite. TheCartesian 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 awell-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 thepower set of its power set is aDedekind-infinite set, having a proper subsetequinumerous 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 awell-orderable set, then it has many well-orderings which are non-isomorphic.
Important ideas discussed by David Burton in his bookThe 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 includeordered sets, cardinality, equivalency,coordinate planes,universal sets, mapping, subsets, continuity, andtranscendence.[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 asπ, integers, andEuler'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 ofThe History of Mathematics: An Introduction, Burton emphasizes how mathematicians such asZermelo,Dedekind,Galileo,Kronecker, Cantor, andBolzano 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]
The set of allintegers, {..., −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 allrational numbers is a countably infinite set as there is a bijection to the set of integers.[3]
The set of allreal numbers is an uncountably infinite set. The set of allirrational numbers is also an uncountably infinite set.[3]
The set of all subsets of the integers is uncountably infinite.
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