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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. |
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. |
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Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set, and this is also equal to the cardinal itself being infinite under the definition of cardinals as initial ordinal. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. |
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Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. |
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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 |
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 [[disjoint union]] \(S\sqcup S\). |
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* It is in bijection with the [[Cartesian product]] \(S\times S\). |
* It is in bijection with the [[Cartesian product]] \(S\times S\). |
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* \(S\) has a countable subset. |
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Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. |
Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. |
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Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of |
Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of \(\omega\), which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. A model of this theory is provided by the set of [[hereditarily finite set]]s. |
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Important kinds of infinite sets include [[countable]] and [[uncountable]] sets. |
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== Properties == |
== Properties == |
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* The set difference of an infinite set and a finite set is infinite. |
* The set difference of an infinite set and a finite set is infinite. |
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* 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 [[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]]. |
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* The [[ |
* The [[Cardinal arithmetic|sum]], product, or exponentiation of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above. |
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== Infinity == |
== Infinity == |
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