Ordering Principle

Theorem
Let $S$ be a set.

Then there exists a total ordering on $S$.

Remarks
As shown in Proof 2 the Ordering Principle is weaker than the Order-Extension Principle (OE).

It is known that it is in fact strictly weaker than OE.

However, the Ordering Principle cannot be proved in Zermel-Fraenkel set theory without the Axiom of Choice.

In fact it is known to be strictly stronger than the Axiom of Choice for Finite Sets.