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

, an infinite set is a set that is not a finite set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. 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 Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

or uncountable
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 ...

.
Properties

The set ofnatural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

(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 that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

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
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 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 , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

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 , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

'' an infinite set is infinite. The Cartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

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
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). ...

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 Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s is an uncountably infinite set. The set of all irrational numbers
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...

References

External links

A Crash Course in the Mathematics Of Infinite Sets

{{Mathematical logic Cardinal numbers