Infinite: Difference between revisions
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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
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* 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 ==
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* 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 ==
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