Editing Infinite set (section)

==Properties==
The set of [[natural numbers]] (whose existence is postulated by the [[axiom of infinity]]) is infinite.<ref name=Bagaria>{{Citation|last=Bagaria|first=Joan|title=Set Theory|date=2019|url=https://plato.stanford.edu/archives/fall2019/entries/set-theory/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-11-30}}</ref> It is the only set that is directly required by the [[axiom]]s 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.<ref>{{cite book |title=Logic, Logic, and Logic |last = Boolos | first = George |edition=illustrated |publisher=Harvard University Press |year=1998 |isbn=978-0-674-53766-8 |page=262 |url=https://books.google.com/books?id=4OPWAAAAMAAJ}}</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.

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.<ref>{{citation
 | last = Boolos | first = George
 | contribution = The advantages of honest toil over theft
 | mr = 1373892
 | pages = 27–44
 | publisher = Oxford Univ. Press, New York
 | series = Logic Comput. Philos.
 | title = Mathematics and mind (Amherst, MA, 1991)
 | year = 1994}}. See in particular [https://books.google.com/books?id=g2OeRwbgFyMC&pg=PA32 pp. 32–33].</ref> 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.