Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element/Proof 2

Theorem

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

Let $x$ be a fixed point of $g$.

Then $x$ is the greatest element of $M$.

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 Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element.

$\blacksquare$

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 $4.1$