Zorn's lemma said to be the same as axiom of choice
Hausdorff maximal principle
Axiom of choice
Well-ordering theorem.A well-known joke alluding to this equivalency (which may defy human intuition) is attributed to Jerry Bona:
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"Zorn's lemma is also equivalent to the strong completeness theorem of first-order logic.Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,