 Zermelo set theory has difficulties with the cumulative hierarchy. The usual form of the Zermelo axioms (or Zermelo’s original form) does not prove the existence of \(V_{\alpha}\) as a set unless \(\alpha\) is finite. If the Axiom of Infinity is reformulated to assert the existence of \(V_{\omega}\), then the ranks proved to exist as sets by Zermelo set theory are exactly those which appear in the natural model \(V_{\omega +\omega}\) of this theory. Also, Zermelo set theory does not prove the existence of transitive closures of sets, which makes it difficult to assign ranks to sets in general. Zermelo set theory plus the assertion that every set belongs to a rank \(V_{\alpha}\) which is a set implies Foundation, the existence of expected ranks \(V_{\alpha}\) (not the existence of such ranks for all ordinals \(\alpha\) but the existence of such a rank containing each set which can be shown to exist), and the existence of transitive closures, and can be interpreted in Zermelo set theory without additional assumptions.