Properties of Class of All Ordinals/Union of Chain of Ordinals is Ordinal

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:

$C$ is a chain of elements of $\On$

$\bigcup C$ is the union of $C$.

$\blacksquare$

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 $1.2 \ (3)$