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{{DISPLAYTITLE:\(\omega\)}}
The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with [[Uncountable|\(\Omega\)]], a common notation for a much larger ordinal. The existence of \(\omega\) is guaranteed by the [[axiom of infinity]].
==Properties==
* It is the first [[infinite]] ordinal.
* It is the first [[limit ordinal]].
* It is considered by some to be the first [[admissible ordinal]].
* Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]].
* It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property.
* It is the next ordinal after [[0]] that isn't a [[successor ordinal]].
* It is additively, multiplicatively, and exponentially [[principal]].
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