# Principle of Finite Induction/Peano Structure

## Theorem

Let $\struct {P, s, 0}$ be a Peano structure.

Let $S \subseteq P$.

Suppose that:

$(1): \quad 0 \in S$
$(2): \quad \forall n: n \in S \implies \map s n \in S$

Then:

$S = P$

## Proof

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

$\blacksquare$