Principle of Finite Induction/Peano Structure

Theorem
Let $\left({P, 0, s}\right)$ be a Peano structure.

Let $S \subseteq P$.

Suppose that:


 * $(1): \quad 0 \in S$
 * $(2): \quad \forall n: n \in S \implies s \left({n}\right) \in S$

Then:


 * $S = P$

Proof
This is nothing but a reformulation of Axiom $(P5)$ of the Peano Axioms.