The following definition shows that not all ordinal numbers are the successors of other ordinal numbers.

As an example, the minimal inductive set $\omega$ is by construction transitive and all of its elements are transitive. Therefore it is an ordinal by the proposition about equivalent notions of ordinals. But there is no ordinal number $\alpha$ such that $s(\alpha)=\omega.$

Definition: Limit Ordinal

A limit ordinal is an ordinal, which is not a successor of any other ordinal, i.e. if \[\forall\alpha\in\Omega:~\gamma\neq\alpha\cup\{\alpha\}.\]

Definitions: 1
Explanations: 2

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011