Sandwich Principle for Slowly Progressing Mapping/Proof
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Theorem
Let $M$ be a class.
Let $g: M \to M$ be a slowly progressing mapping on $M$.
Let $M$ be a minimally inductive class under $g$.
Then $x \subsetneqq y \subsetneqq \map g x$ can never hold.
Proof
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Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 9$ Supplement -- optional: Exercise $9.1$