# Definition:Ordinal Sequence

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## 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