User:Dfeuer/Principle of Recursive Definition/Peano Numbers

Theorem

Let $\N$ be a class satisfying Peano's axioms.

Let $A$ be a set or class.

Let $F: A \to A$ be a mapping.

Let $a \in A$.

Then there is a unique mapping $G: \N \to A$ such that:

$\map G 0 = a$

$\forall n \in \N: \map G {n^+} = \map F {\map G n}$

Proof

Finite Sections

For each $n \in \N$, there is a unique mapping $g_n: \N_n \to A$ such that: