G-Tower is Well-Ordered under Subset Relation/Corollary

Corollary to $g$-Tower is Well-Ordered under Subset Relation

Let $M$ be a class.

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

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

Let $y \in M$ other than $\O$.

Then the strict lower closure $y^\subset$ of $y$ is non-empty and exactly one of the following conditions holds:

$(\text C 1): \quad y^\subset$ has a greatest element $x$ and $\map g x = y$

$(\text C 2): \quad y^\subset$ has no greatest element $x$ and $\bigcup y^\subset = y$

Proof

We have by hypothesis $y \ne \O$.

Then $y$ is not the smallest element of $M$.

Hence $\O \in y^\subset$ and $y^\subset$ is non-empty.

The result follows from $g$-Tower is Well-Ordered under Subset Relation, taking $A = \set y$.

The smallest element of $\set y$ is of course $y$ itself.

$\blacksquare$

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: $\S 3$ The well ordering of $g$-towers: Corollary $3.2$