- Successor ordinal
When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a

**successor operation**"S" to get the next higher one. Using von Neumann's ordinal numbers (the standard ordinals used in set theory), we have, for any ordinal number,:$S(alpha)\; =\; alpha\; cup\; \{alpha\}.$

Since the ordering on the ordinal numbers α < β if and only if $alpha\; in\; eta$, it is immediate that there is no ordinal number between α and "S"(α) and it is also clear that α < "S"(α). An ordinal number which is "S"(β) for some ordinal β, or equivalently, an ordinal with a maximum element, is called a

**successor ordinal**. Ordinals which are neither zero nor successors are called**limit ordinals**. We can use this operation to define ordinal addition rigorously via transfinite recursion as follows::$alpha\; +\; 0\; =\; alpha!$:$alpha\; +\; S(eta)\; =\; S(alpha\; +\; eta)!$

and for a limit ordinal λ

:$alpha\; +\; lambda\; =\; igcup\_\{eta\; lambda\}\; (alpha\; +\; eta)$

In particular, "S"(α) = α + 1. Multiplication and exponentiation are defined similarly.

The successor points and zero are the

isolated point s of the class of ordinal numbers, with respect to theorder topology .**ee also***

ordinal arithmetic

*limit ordinal

*successor cardinal

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