Well-Ordering Principle

Theorem
Every non-empty subset of $$\mathbb{N}$$ has a minimal or "smallest" element.

This is called the well-ordering property of $$\mathbb{N}$$, or the well-ordering principle.

The well-ordering principle also holds for $$\mathbb{N}^*$$.

Proof

 * The set of natural numbers is defined as the archetype of the naturally ordered semigroup.

From the definition of the naturally ordered semigroup, $$\left({S, \circ; \preceq}\right)$$ is well-ordered by $$\preceq$$.

So as $$\left({\mathbb{N}, +; \le}\right) \cong \left({S, \circ; \preceq}\right)$$ the result follows.


 * As $$\mathbb{N}^* = \mathbb{N} - \left\{{0}\right\}$$, by Set Difference Subset $$\mathbb{N}^* \subseteq \mathbb{N}$$.

As $$\mathbb{N}$$ is well-ordered, by definition, every subset of $$\mathbb{N}$$ has a minimal element.