In set theory
, an infinite set is a set
that is not a finite set
sets may be countable
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 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
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.
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
[. See in particula]
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.
Countably infinite sets
The set of all integer
s, 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 number
s is an uncountably infinite set. The set of all irrational numbers
is also an uncountably infinite set.
* Aleph number
External links A Crash Course in the Mathematics Of Infinite Sets