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Successor OrdinalWhen 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.
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