Ordering Principle

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Theorem

Let $S$ be a set.


Then there exists a total ordering on $S$.


Proof 1

From the 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


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.