- 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,
Since the ordering on the ordinal numbers α < β if and only if , it is immediate that there is no ordinal number between α and "S"(α) and it is also clear that α < "S"(α). An ordinal number which is "S"(β) for some ordinal β, or equivalently, an ordinal with a maximum element, is called a successor ordinal. Ordinals which are neither zero nor successors are called limit ordinals. We can use this operation to define ordinal addition rigorously via transfinite recursion as follows:
and for a limit ordinal λ
In particular, "S"(α) = α + 1. Multiplication and exponentiation are defined similarly.
The successor points and zero are the
isolated points of the class of ordinal numbers, with respect to the order topology.
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