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.
$\blacksquare$
Sources
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): well ordered