Successor ordinal - Definition

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,

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

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 <math>\alpha \in \beta<math>, 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:

<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, S(α) = α + 1. Multiplication and exponentiation are defined similarly. Also see limit ordinal.