Well-Ordering Principle

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

That is, the relational structure $\struct {\N, \le}$ on the set of natural numbers $\N$ under the usual ordering $\le$ forms a well-ordered set.

This is called the well-ordering principle.

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

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 Zermelo's 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