There are famously-many other axioms that can be added to ZF, and which turn out to be equivalent to Choice.

Of the following three equivalent extensions to ZF:

  * Axiom of Choice

  * Zorn's Lemma

  * Well-ordering Principle

Jerry Bona quipped:

"The Axiom of Choice is obviously true, the Well-ordering Principle obviously false, and who can tell about Zorn's Lemma?"