Relation on Slowly Progressing Mapping which fulfils conditions of General Double Induction Principle

Theorem

Let $M$ be a class.

Let $g: M \to M$ be a slowly progressing mapping on $M$.

Let $\map \RR {x, y}$ be the relation on $M$ defined as:

$\forall x, y \in M: \tuple {x, y} \in \R \iff x \subseteq y \lor y \subseteq x$

Then $\RR$ satisfies the conditions of the General Double Induction Principle:

$({\text D'}_1): \quad \map \RR {x, 0}$ and $\map \RR {0, x}$ hold for every $x \in M$

$({\text D'}_2): \quad \forall x, y \in M: \paren {\map \RR {x, y} \land \map \RR {x, \map g y} \land \map \RR {\map g x, y} } \implies \map \RR {\map g x, \map g y}$

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: Lemma $S_2$