Editing Successor ordinal

{{Short description|Operation on ordinal numbers}}
In [[set theory]], the '''successor''' of an [[ordinal number]] ''α'' is the smallest ordinal number greater than&nbsp;''α''.  An ordinal number that is a successor is called a '''successor ordinal'''. The ordinals 1, 2, and 3 are the first three successor ordinals and the ordinals ω+1, ω+2 and ω+3 are the first three infinite successor ordinals.
 
==Properties==
Every ordinal other than 0 is either a successor ordinal or a [[limit ordinal]].<ref name="cameron">{{citation|title=Sets, Logic and Categories|series=Springer Undergraduate Mathematics Series|first=Peter J.|last=Cameron|publisher=Springer|year=1999|isbn=9781852330569|page=46|url=https://books.google.com/books?id=sDfdbBQ75MQC&pg=PA46}}.</ref>

==In Von Neumann's model==
Using [[Ordinal number#Von Neumann definition of ordinals|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<ref name="cameron"/>

:<math>S(\alpha) = \alpha \cup \{\alpha\}.</math>

Since the ordering on the ordinal numbers is given by ''α''&nbsp;<&nbsp;''β'' if and only if ''α''&nbsp;∈&nbsp;''β'', it is immediate that there is no ordinal number between α and ''S''(''α''), and it is also clear that ''α''&nbsp;<&nbsp;''S''(''α'').

==Ordinal addition==

The successor operation can be used to define [[ordinal arithmetic|ordinal addition]] rigorously via [[transfinite induction|transfinite recursion]] as follows:

:<math>\alpha + 0 = \alpha\!</math>
:<math>\alpha + S(\beta) = S(\alpha + \beta)</math>

and for a limit ordinal ''λ''

:<math>\alpha + \lambda = \bigcup_{\beta < \lambda} (\alpha + \beta)</math>

In particular, {{nowrap|1=''S''(''α'') = ''α'' + 1}}. Multiplication and exponentiation are defined similarly.

==Topology==
The successor points and zero are the [[isolated point]]s of the class of ordinal numbers, with respect to the [[order topology]].<ref>{{citation|title=The Joy of Sets: Fundamentals of Contemporary Set Theory|series=[[Undergraduate Texts in Mathematics]]|first=Keith|last=Devlin|publisher=Springer|year=1993|isbn=9780387940946|at=Exercise 3C, p.&nbsp;100|url=https://books.google.com/books?id=hCv-vFu4jskC&pg=PA100}}.</ref>

==See also==
*[[Ordinal arithmetic]]
*[[Limit ordinal]]
*[[Successor cardinal]]

==References==
{{reflist}}

{{DEFAULTSORT:Successor Ordinal}}
[[Category:Ordinal numbers]]