Well-Ordering Principle

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

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

Some sources give this as the least-integer principle.

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

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


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

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

Also see
Some authors extend the scope of this theorem to include:
 * Integers Bounded Below has Minimal Element
 * Integers Bounded Above has Maximal Element

This theorem should not be confused with the Well-Ordering Theorem, which states that any set can have an ordering under which that set is a well-ordered set.