# Successor ordinal

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**.

## Contents

## PropertiesEdit

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

## In Von Neumann's modelEdit

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 additionEdit

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.

## TopologyEdit

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

## See alsoEdit

## ReferencesEdit

- ^
^{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.