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 $\On$.

Length

Let $\alpha$ be an ordinal.

Let $\theta$ be an ordinal sequence whose domain is $\alpha$.

Then $\alpha$ can be referred to as the length of $\theta$.

The length of $\theta$ can be denoted $\size \theta$.

Also see

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

• Results about ordinal sequences can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems