Well-Ordering is not necessarily Usual Ordering

Theorem

Let $S$ be a set of numbers.

According to Zermelo's Well-Ordering Theorem, $S$ can be well-ordered.

However, the usual ordering on $S$ may not necessarily be a well-ordering.

Proof

From Rational Numbers are Well-Orderable, it is possible to apply a well-ordering to the set of rational numbers $\Q$.

However, the usual ordering on $\Q$ is not a well-ordering.

Indeed:

$\set {x \in \Q: x \le 0}$

has no smallest element.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Discussion