In

pp. 32–33

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.

A Crash Course in the Mathematics Of Infinite Sets

{{Mathematical logic Cardinal numbers

set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...

, an infinite set is a set that is not a finite set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. Infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (band), a South Korean boy band
*''Infinite'' (EP), debut EP of American musi ...

sets may be countable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

or uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many Element (mathematics), elements to be countable set, countable. The uncountability of a set is closely related to its cardinal number: a set ...

.
Properties

The set ofnatural numbers
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

(whose existence is postulated by the axiom of infinity
In axiomatic set theory and the branches of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

) is infinite. It is the only set that is directly required by the axiom
An axiom, postulate, or assumption is a Statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek () 'that which is thought worthy or ...

s to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...

(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
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are u ...

whose cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is that natural number.
If the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the ax ...

holds, then a set is infinite if and only if it includes a countable infinite subset.
If a set of setsIn set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any typ ...

is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

'' an infinite set is infinite. The Cartesian product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, 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
Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion.
In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...

of its power set is a Dedekind-infinite set, having a proper subset equinumerous
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

to itself.. See in particulapp. 32–33

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.

Examples

Countably infinite sets

The set of allinteger
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

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 allreal number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s is an uncountably infinite set. The set of all irrational numbers
In mathematics, the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...

is also an uncountably infinite set.
See also

*Aleph number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

*Cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

*Ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another.
Any finite collection of ob ...

References

External links

A Crash Course in the Mathematics Of Infinite Sets

{{Mathematical logic Cardinal numbers