Types of ordinals

Let be an Ordinal.
We consider two cases,
depending on whether has a biggest element or not.

Case 1

has a greatest element .
Then so .
In this case we call a successor ordinal.
Note that

Case 2

has no greatest element.
So if ,
i.e. ,
then there is a ,
i.e. .
It follows that .
We say is a limit ordinal.