# GetWiki

*infinite set*

ARTICLE SUBJECTS

being →

database →

ethics →

fiction →

history →

internet →

language →

linux →

logic →

method →

news →

policy →

purpose →

religion →

science →

software →

truth →

unix →

wiki →

ARTICLE TYPES

essay →

feed →

help →

system →

wiki →

ARTICLE ORIGINS

critical →

forked →

imported →

original →

infinite set

[ temporary import ]

**please note:**

- the content below is remote from Wikipedia

- it has been imported raw for GetWiki

**infinite set**is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:

- the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
- the set of all real numbers is an uncountably infinite set.

## Properties

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 axioms to be infinite. The existence of any other infinite set can be proved in Zermeloâ€“Fraenkel set theory (ZFC) 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.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. Any 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 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.{{citation
| last = Boolos | first = George

| contribution = The advantages of honest toil over theft

| mr = 1373892

| pages = 27â€“44

| publisher = Oxford Univ. Press, New York

| series = Logic Comput. Philos.

| title = Mathematics and mind (Amherst, MA, 1991)

| year = 1994}}. See in particular pp. 32â€“33. If the axiom of choice is also true, 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.| contribution = The advantages of honest toil over theft

| mr = 1373892

| pages = 27â€“44

| publisher = Oxford Univ. Press, New York

| series = Logic Comput. Philos.

| title = Mathematics and mind (Amherst, MA, 1991)

| year = 1994}}. See in particular pp. 32â€“33. If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.

## See also

## References

{{Reflist}}## External links

- {{MathWorld |title=Infinite Set |id=InfiniteSet }}

**- content above as imported from Wikipedia**

- "

- time: 4:13am EDT - Mon, Apr 22 2019

- "

__infinite set__" does not exist on GetWiki (yet)- time: 4:13am EDT - Mon, Apr 22 2019

[ this remote article is provided by Wikipedia ]

LATEST EDITS [ see all ]

GETWIKI 09 MAY 2016

GETWIKI 18 OCT 2015

GETWIKI 20 AUG 2014

GETWIKI 19 AUG 2014

GETWIKI 18 AUG 2014

© 2019 M.R.M. PARROTT | ALL RIGHTS RESERVED