Minimally Inductive Class under Progressing Mapping induces Nest/Proof 2
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Theorem
Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Then $M$ is a nest in which:
- $\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Proof
A minimally inductive class under $g$ is the same thing as a minimally closed class under $g$ with respect to $\O$.
The result then follows by a direct application of Minimally Closed Class under Progressing Mapping induces Nest.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Exercise $4.1$