 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.

## 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

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

## 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 ?

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

## See also

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