Characteristics of Minimally Inductive Class under Progressing Mapping

Theorem

Let $M$ be a class which is minimally inductive under a progressing mapping $g$.

Then for all $x, y \in M$:

Sandwich Principle

$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$

Image of Proper Subset is Subset

$x \subset y \implies \map g x \subseteq y$

Mapping Preserves Subsets

$x \subseteq y \implies \map g x \subseteq \map g y$

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: Theorem $4.10$