Ackermann introduced in [1] a system of axiomatic set theory. The quantifiers of this set theory range over a universe of objects which we call classes. Among the classes we distinguish the sets. Here we shall show that, in some sense, all the theorems of Ackermann's set theory can be proved in Zermelo-Fraenkel's set theory. We shall also show that, on the other hand, it is possible to prove in Ackermann's set theory very strong theorems of the Zermelo-Fraenkel set theory.