Union of Chain of Ordinal Sequences is Ordinal Sequence

Theorem

Let $C$ be a chain of ordinal sequences.

Then $\bigcup C$ is an ordinal sequence.

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: Lemma $5.2 \ (1)$