General Double Induction Principle/Proof
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Theorem
Let $M$ be a class.
Let $g: M \to M$ be a mapping on $M$.
Let $M$ be a minimally inductive class under $g$.
Let $\RR$ be a relation which satisfies the following conditions:
\(({\text D'}_1)\) | $:$ | \(\ds \forall x \in M:\) | \(\ds \map \RR {x, 0} \land \map \RR {0, x} \) | ||||||
\(({\text D'}_2)\) | $:$ | \(\ds \forall x, y \in M:\) | \(\ds \paren {\map \RR {x, y} \land \map \RR {x, \map g y} \land \map \RR {\map g x, y} } \) | \(\ds \implies \) | \(\ds \map \RR {\map g x, \map g y} \) |
Then:
- $\forall x, y \in M: \map \RR {x, y}$
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: Exercise $9.1$