 Naturally, we may also want to assume the Well-ordering Principle which states that there is a binary relation which is a total order on the individuals and satisfies the condition that every non-empty set of individuals has a least element in the order. It is easy to see that this principle implies (AC′). In ZFC there are hundreds of equivalent formulations of the Axiom of Choice or weaker versions of it. Most of the formulations that are equivalent in ZFC are non-equivalent in second-order logic. This is because set theory allows free movement between sets and their power sets, but second-order logic does  not[11].  To obtain similar equivalence proofs as in set theory, one would have to move between second-order logic and third order logic, see Gaßner (1994).