Hausdorff's Maximal Principle is equivalent to Axiom of Choice

Theorem

Every ordered set has a maximal chain if and only if the axiom of choice holds.

Proof

• Axiom of Choice implies Hausdorff's Maximal Principle

• Hausdorff's Maximal Principle implies Zermelo's Well-Ordering Theorem
• Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 16$: Zorn's Lemma: Exercise $\text{(i)}$