Successor Ordinal
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Successor Ordinal

In set theory, the successor of an ordinal number ? is the smallest ordinal number greater than ?. 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(?) of an ordinal number ? is given by the formula[1]

${\displaystyle S(\alpha )=\alpha \cup \{\alpha \}.}$

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

The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows:

${\displaystyle \alpha +0=\alpha \!}$
${\displaystyle \alpha +S(\beta )=S(\alpha +\beta )}$

and for a limit ordinal ?

${\displaystyle \alpha +\lambda =\bigcup _{\beta <\lambda }(\alpha +\beta )}$

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.[2]

## References

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