Principle of Finite Induction/Peano Structure
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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.
$\blacksquare$