Properties of Mapping on Class of All Ordinals

Theorem

Let $\On$ denote the class of all ordinals.

Let $F$ be a mapping on $\On$.

For $\alpha \in \On$, let $F \restriction \alpha$ denote the restriction of $F$ to $\alpha$.

Then:

\((1)\)	$:$					\(\ds F \restriction 0 \)	\(\ds = \)	\(\ds 0 \)
\((2)\)	$:$		For a given limit ordinal $\lambda$:			\(\ds F \restriction \lambda \)	\(\ds = \)	\(\ds \bigcup_{\alpha \mathop < \lambda} F \restriction \alpha \)

Proof

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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: Remark