Union of Slow g-Tower is Well-Orderable

Theorem

Let $M$ be a class.

Let $g: M \to M$ be a slowly progressing mapping on $M$.

Let $M$ be a slow $g$-tower.

Then $\ds \bigcup M$ is well-orderable.

Proof

Let $M$ be a slow $g$-tower.

From Slow $g$-Tower is Slowly Well-Ordered under Subset Relation we have that $M$ is slowly well-ordered under $\subseteq$.

The result follows from Union of Slowly Well-Ordered Class under Subset Relation is Well-Orderable.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 4$ Well ordering and choice: Theorem $4.5$