In set theory, the **successor** of an ordinal number *α* is the smallest ordinal number greater than *α*. An ordinal number that is a successor is called a **successor ordinal**.

## Properties

Every ordinal other than 0 is either a successor ordinal or a limit ordinal.^{[1]}

## 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^{[1]}

Since the ordering on the ordinal numbers is given by *α* < *β* if and only if *α* ∈ *β*, it is immediate that there is no ordinal number between α and *S*(*α*), and it is also clear that *α* < *S*(*α*).

## Ordinal addition

The successor operation can be used 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.

## Topology

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology.^{[2]}

## See also

## References

- ^
^{a}^{b}Cameron, Peter J. (1999),*Sets, Logic and Categories*, Springer Undergraduate Mathematics Series, Springer, p. 46, ISBN 9781852330569. **^**Devlin, Keith (1993),*The Joy of Sets: Fundamentals of Contemporary Set Theory*, Undergraduate Texts in Mathematics, Springer, Exercise 3C, p. 100, ISBN 9780387940946.