G-Tower is Well-Ordered under Subset Relation/Corollary
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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 $3.2$