Infinite: Difference between revisions
no edit summary
OfficialURL (talk | contribs) (more info) |
RhubarbJayde (talk | contribs) No edit summary |
||
(3 intermediate revisions by one other user not shown) | |||
Line 1:
A [[set]] is said to be '''infinite''' when it is not [[finite]]. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any. Any [[countable]] set is infinite, but there are non-countable infinite sets. An [[ordinal]] is called infinite when it is the [[order type]] of an infinite [[well-ordered set]]. Under the von Neumann representation, this is just equivalent to the ordinal itself being infinite, as a set. The smallest infinite ordinal is [[omega|\(\omega\)]]. Every ordinal larger than it is infinite, and every ordinal smaller than it is finite.
There are various equivalent ways to phrase the definition of an infinite set. Assuming the [[axiom of choice]], a set \(S\) is infinite if and only if
Line 9 ⟶ 7:
* It is in bijection with the [[disjoint union]] \(S\sqcup S\).
* It is in bijection with the [[Cartesian product]] \(S\times S\).
* \(S\) has a countable subset.
Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''.
Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of
== Properties ==
Line 20 ⟶ 17:
* The set difference of an infinite set and a finite set is infinite.
* The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by [[0]], and exponentiation with base 0 or [[1]].
* The [[
== Infinity ==
A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts. It can be used purely notationally, such as when denoting [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits to infinity], [https://en.wikipedia.org/wiki/Series_(mathematics) series], and [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an object in a structure such as the [https://en.wikipedia.org/wiki/Extended_real_number_line extended real numbers]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number
\(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\).
== External links ==
* {{Wikipedia|Infinite set}}▼
* {{Mathworld|Infinite Set}}
* {{Mathworld|Infinity}}
▲* {{Wikipedia|Infinite set}}
* {{Wikipedia|Infinity}}
|