Length of Union of Chain of Ordinal Sequences

Theorem

Let $C$ be a chain of ordinal sequences.

Let $\lambda$ be a limit ordinal such that all elements of $C$ are of length (strictly) less than $\lambda$.

Suppose that for every ordinal $\alpha < \lambda$ there exists $\theta \in C$ such that:

$\size \theta = \alpha$

where $\size \theta$ denotes the length of $\theta$.

Then:

$\size {\bigcup C} = \lambda$

where $\bigcup C$ denotes the union of $C$.

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 $5.2 \ (1)$