Successor ordinal

All you want to know about 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 $\alpha \in \beta$ , 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: