Definition:Extending Operation

Definition

Let $S$ denote the class of all ordinal sequences.

Let $E: S \to S$ be a mapping whose behaviour is such that:

for all $\theta \in S$: if $\theta$ has length $\alpha$, then $\map E \theta$ has length $\alpha^+$

where $\alpha^+$ is the successor ordinal of $\alpha$.

Then $E$ is an extending operation.

That is, $E$ extends every ordinal sequence of length $\alpha$ to an ordinal sequence of length $\alpha^+$.

Also see

• Results about extending operations 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