Properties of Class of All Ordinals/Union of Chain of Ordinals is Ordinal
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Theorem
Let $\On$ denote the class of all ordinals.
Let $C$ be a chain of elements of $\On$.
Then its union $\bigcup C$ is also an element of $\On$.
Proof
We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping.
Hence $\On$ is a fortiori a superinductive class with respect to the successor mapping.
Hence, by definition of superinductive class:
- $\On$ is closed under chain unions.
That is:
- $\forall C \in \On: \bigcup C \in \On$
where:
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.2 \ (3)$