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| {{Unreferenced|date=December 2009}}
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| In [[set theory]], the '''successor''' of an [[ordinal number]] ''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a '''successor ordinal'''. Every ordinal other than 0 is either a successor ordinal or a [[limit ordinal]].
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| Using [[von Neumann]]'s ordinal numbers (the standard model of the ordinals used in set theory), the successor ''S''(''α'') of an ordinal number ''α'' is given by the formula
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| :<math>S(\alpha) = \alpha \cup \{\alpha\}.</math>
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| Since the ordering on the ordinal numbers α < β if and only if ''α'' ∈ ''β'', it is immediate that there is no ordinal number between α and ''S''(α), and it is also clear that α < ''S''(α).
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| The successor operation can be used to define [[ordinal arithmetic|ordinal addition]] rigorously via [[transfinite induction|transfinite recursion]] as follows:
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| :<math>\alpha + 0 = \alpha\!</math>
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| :<math>\alpha + S(\beta) = S(\alpha + \beta)\!</math>
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| and for a limit ordinal λ
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| :<math>\alpha + \lambda = \bigcup_{\beta < \lambda} (\alpha + \beta)</math>
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| In particular, ''S''(α) = α + 1. Multiplication and exponentiation are defined similarly.
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| The successor points and zero are the [[isolated point]]s of the class of ordinal numbers, with respect to the [[order topology]].
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| ==See also==
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| *[[ordinal arithmetic]]
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| *[[limit ordinal]]
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| *[[successor cardinal]]
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| {{DEFAULTSORT:Successor Ordinal}}
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| [[Category:Ordinal numbers]]
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Latest revision as of 05:25, 6 June 2014
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