Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice

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Theorem

Zermelo's Well-Ordering Theorem holds if and only if the Axiom of Choice holds.


That is, every set is well-orderable if and only if every collection of sets has a choice function.


Proof

Necessary Condition

Suppose the Axiom of Choice holds.

Then Zermelo's Well-Ordering Theorem holds by Zermelo's Well-Ordering Theorem itself.

That is, every set is well-orderable.

$\Box$


Sufficient Condition

Let it be supposed that every set is well-orderable.

Then by the converse to Zermelo's Well-Ordering Theorem:

the Axiom of Choice holds.

$\blacksquare$


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