Principle of Finite Induction/Zero-Based

Theorem
Let $S \subseteq \N$ be a subset of the natural numbers.

Suppose that:


 * $(1): \quad 0 \in S$


 * $(2): \quad \forall n \in \N : n \in S \implies n + 1 \in S$

Then:


 * $S = \N$

Proof
Consider $\N$ defined as a naturally ordered semigroup.

The result follows directly from Principle of Mathematical Induction for Naturally Ordered Semigroup.