Ordering Principle/Proof 1

Theorem

Let $S$ be a set.

Then there exists a total ordering on $S$.

Proof

From Zermelo's Well-Ordering Theorem, $S$ has a well-ordering.

The result follows from Well-Ordering is Total Ordering.

$\blacksquare$

Axiom of Choice

This theorem depends on the Axiom of Choice, by way of Zermelo's Well-Ordering Theorem.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.