Ordering Principle
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Theorem
Let $S$ be a set.
Then there exists a total ordering on $S$.
Proof 1
From Zermelo's Well-Ordering Theorem, $S$ has a well-ordering.
The result follows from Well-Ordering is Total Ordering.
$\blacksquare$
Proof 2
This theorem follows trivially from the Order-Extension Principle.
$\blacksquare$
Remarks
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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.