Well-Ordering Principle/Proof by Restriction of Real Numbers

Theorem

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.

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 $S$.

$\blacksquare$