# Successor ordinal

When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one. Using von Neumann's ordinal numbers (the standard ordinals used in set theory), we have, for any ordinal number,

[itex]S(\alpha) = \alpha \cup \{\alpha\}.[itex]

It is immediate that there is no ordinal number between α and S(α) and with the ordering on the ordinal numbers α < β if and only if [itex]\alpha \in \beta[itex], it is clear that α < S(α). An ordinal number which is S(β) for some ordinal β is called a successor ordinal. Ordinals which are not successors are called limit ordinals. We can use this operation to define ordinal addition rigorously via transfinite induction as follows:

[itex]\alpha + 0 = \alpha[itex]
[itex]\alpha + S(\beta) = S(\alpha + \beta)[itex]

and for a limit ordinal λ

[itex]\alpha + \lambda = \bigcup_{\beta < \lambda} (\alpha + \beta)[itex]

In particular, S(α) = α + 1. Multiplication and exponentiation are defined similarly. Also see limit ordinal.zh:后继序数

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