Well-Ordering Principle

Theorem
Every non-empty subset of $\N$ has a smallest (or first) element.

This is called the well-ordering principle.

The well-ordering principle also holds for $\N_{\ne 0}$.

Proof
Consider the natural numbers $\N$ defined as the naturally ordered semigroup $\struct {S, \circ, \preceq}$.

From its definition, $\struct {S, \circ, \preceq}$ is well-ordered by $\preceq$.

The result follows.

As $\N_{\ne 0} = \N \setminus \set 0$, by Set Difference is Subset $\N_{\ne 0} \subseteq \N$.

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

Also known as
This is otherwise known as:
 * the well-ordering property (of $\N$)
 * the least-integer principle
 * the principle of the least element.

Note that some authors cite this as the well-ordering theorem.

However, this allows it to be confused even more easily with the Well-Ordering Theorem, which states that any set can have an ordering under which that set is a well-ordered set.

Also see

 * Principle of Mathematical Induction
 * Second Principle of Mathematical Induction


 * Equivalence of Well-Ordering Principle and Induction

Some authors extend the scope of this theorem to include:


 * Set of Integers Bounded Below has Smallest Element
 * Set of Integers Bounded Above has Greatest Element