Successor ordinal

Encyclopedia : S : SU : SUC : 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,

[S(\alpha) = \alpha \cup \.]

Since the ordering on the ordinal numbers α < β if and only if [\alpha \in \beta], 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 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:

[\alpha + 0 = \alpha]

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

and for a limit ordinal λ

[\alpha + \lambda = \bigcup_ (\alpha + \beta)]

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

Successor ordinal

See also