Strict Lower Closure of G-Tower is Set of Elements which are Proper Subsets
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Theorem
Let $M$ be a class.
Let $g: M \to M$ be a progressing mapping on $M$.
Let $M$ be a $g$-tower.
Let $x \in M$.
Then the strict lower closure of $x$ is the set of all elements of $M$ that are proper subsets of $M$.
Proof
Follows directly from the definitions.
$\blacksquare$
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