 The axiom scheme of Foundation was added as a definite conception of what the universe of sets is like. The idea of the cumulative hierarchy of sets is that we construct sets in a sequence of stages indexed by the ordinals: at stage 0, the empty set is constructed; at stage \(\alpha + 1\), all subsets of the set of stage \(\alpha\) sets are constructed; at a limit stage \(\lambda\), the union of all stages with index less than \(\lambda\) is constructed. Replacement is important for the implementation of this idea, as Z only permits one to construct sets belonging to the stages \(V_n\) and \(V_{\omega +n}\) for \(n\) a natural number (we use the notation \(V_{\alpha}\) for the collection of all sets constructed at stage \(\alpha)\). The intention of the Foundation Axiom is to assert that every set belongs to some \(V_{\alpha}\) ; the commonest formulation is the mysterious assertion that for any nonempty set \(A\), there is an element \(x\) of \(A\) such that \(x\) is disjoint from \(A\). To see that this is at least implied by Foundation, consider that there must be a smallest \(\alpha\) such that \(A\) meets \(V_{\alpha}\), and any \(x\) in this \(V_{\alpha}\) will have elements (if any) only of smaller rank and so not in \(A\).