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 $s(\alpha)=\omega.$
A limit ordinal is an ordinal, which is not a successor of any other ordinal, i.e. if \[\forall\alpha\in\Omega:~\gamma\neq\alpha\cup\{\alpha\}.\]
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Explanations: 2