 The axiom scheme of Replacement was added by Fraenkel to make it possible to construct larger sets (even \(\aleph_{\omega}\) cannot be proved to exist in Zermelo set theory). The basic idea is that any collection the same size as a set is a set, which can be logically formulated as follows: if \(\phi(x,y)\) is a functional formula \(\forall x\forall y\forall z[(\phi(x,y) \amp \phi(x,z)) \rightarrow y = z\)] and \(A\) is a set then there is a set \(\{y \mid \exists x \in A(\phi(x,y))\}\).