It turned out, though, that the

*well*-*ordering*theorem is equivalent to the*axiom**of**choice*, in the sense that either one together with the Zermelo–Fraenkel axioms is sufficient to prove the other, in first order logic (the same applies to Zorn's Lemma). In second order logic, however, the*well*-*ordering*theorem is strictly stronger than the*axiom**of**choice*: from the*well*-*ordering*theorem one may deduce the*axiom**of**choice*, but from the*axiom**of**choice*one cannot deduce the*well*-*ordering*theorem.