Union of g-Tower is Greatest Element and Unique Fixed Point
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Theorem
Let $M$ be a set.
Let $g: M \to M$ be a progressing mapping on $M$.
Let $M$ be a $g$-tower.
Then:
- $\ds \bigcup M \in M$
- $\ds \bigcup M$ is the greatest element (under the subset relation) of $M$
- $\ds \bigcup M$ is the unique fixed point of $M$.
Proof
Let the hypothesis be assumed.
By definition, $M$ is a nest which is also a set
Hence by definition $M$ is a chain.
By $g$-Tower is Closed under Chain Unions:
- $\ds \bigcup M \in M$
It follows directly that $\ds \bigcup M$ is the greatest element (under the subset relation) of $M$
Hence $\ds \bigcup M$ cannot be a proper subset of $\ds \map g {\bigcup M}$.
Thus:
- $\ds \bigcup M = \map g {\bigcup M}$
and the result follows.
$\blacksquare$
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: $\S 2$ Superinduction and double superinduction: Theorem $2.6$