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,

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 , 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:

and for a limit ordinal λ

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