# 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 $\N_{\ne 0}$:

Let $\le_1$ denote the restriction of the usual ordering $\le$ to the set $\N_{\ne 0}$ of natural numbers without zero.

The relational structure $\struct {\N_{\ne 0}, \le_1}$ forms a well-ordered set.

## Proof using Naturally Ordered Semigroup

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.

$\blacksquare$

## Proof using Von Neumann Construction

From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.

The result is a direct application of Minimally Inductive Class under Progressing Mapping is Well-Ordered under Inclusion.

$\blacksquare$

## Proof by Restriction of Real Numbers

Let $S$ be a non-empty subset of the set of natural numbers $\N$.

We take as axiomatic that $\N$ is itself a subset of the set of real numbers $\R$.

Thus $S \subseteq \R$.

By definition:

- $\forall n \in \N: n \ge 0$

and so:

- $\forall n \in S: n \ge 0$

Hence $0$ is a lower bound of $S$.

This establishes the fact that $S$ is bounded below.

By the Continuum Property, we have that $S$ admits an infimum.

Hence let $b = \inf S$.

Because $b$ is the infimum of $S$, it follows that $b + 1$ is not a lower bound of $S$.

So, for some $n \in S$:

- $n < b + 1$

Aiming for a contradiction, suppose $n$ is not the smallest element of $S$.

Then there exists $m \in S$ such that:

- $b \le m < n < b + 1$

from which it follows that:

- $0 < n - m < 1$

But there exist no natural numbers $k$ such that $0 < k < 1$.

Hence $n = b$ is the smallest element of $S$.

$\blacksquare$

## 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

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

## Sources

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