Successor ordinal

Properties

Every ordinal other than 0 is either a successor ordinal or a limit ordinal.

In Von Neumann's model

Using von Neumann's ordinal numbers (the standard model of the ordinals used in set theory), the successor S(α) of an ordinal number α is given by the formula

:S(\alpha) = \alpha \cup {\alpha}.

Since the ordering on the ordinal numbers is given by α

Ordinal addition

The successor operation can be used 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_{\beta

In particular, . Multiplication and exponentiation are defined similarly.

Topology

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology.

References

References

1. Cameron, Peter J.. (1999). "Sets, Logic and Categories". Springer.
2. Devlin, Keith. (1993). "The Joy of Sets: Fundamentals of Contemporary Set Theory". Springer.