The Power Set axiom is used to obtain \(V_{\alpha +1}\) from \(V_\alpha\). Replacement and Union allow one to form \(V_\alpha\) for \(\alpha\) a limit ordinal. Indeed, consider the function that assigns to each \(\beta &lt;\alpha\) the set \(V_\beta\). By Replacement, the collection of all the \(V_\beta\), for \(\beta &lt;\alpha\), is a set, hence the Union axiom applied to that set yields \(V_\alpha\). The axiom of Infinity is needed to prove the existence of \(\omega\) and hence of the transfinite sequence of ordinals. Finally, the axiom of Foundation is equivalent, assuming the other axioms, to the statement that every set belongs to some \(V_\alpha\), for some ordinal \(\alpha\). Thus, ZF proves that the set theoretic universe, denoted by \(V\), is the union of all the \(V_\alpha\), \(\alpha\) an ordinal.