Definition:Ordinal Sequence

Definition
An ordinal sequence is a mapping $\theta$ whose domain is an ordinal $\alpha$.

That is, the domain of $\theta$ is the set of all ordinals $\gamma$ such that $\gamma < \alpha$.

Such a sequence can be referred to as an $\alpha$-sequence.

Hence an $\On$-sequence is a mapping whose domain is the class of all ordinals $\On$.

Also see

 * Class of All Ordinals is Ordinal, demonstrating that an $\On$-sequence is still an ordinal sequence