 Zermelo-Fraenkel set theory (ZF) is another theory that builds on the idea of an implicit hierarchy to circumvent the paradoxes. However, it does so in a much less direct way than NF. In ZF, sets are built bottom-up, starting with the empty set and iterating a construction of bigger and bigger sets using the operations of union and power set. This produces a hierarchy with the empty set on the lowest level, level 0, and with the power set operation producing a set of level \(\alpha +1\) from a set of level \(\alpha\). Analogous to Kripke’s iterative construction, the procedure is continued into the transfinite using the union operator at the limit ordinal levels. The hierarchy obtained is called the cumulative hierarchy. One of the axioms of ZF, the axiom of foundation, states that every set of ZF lives on a certain level in this cumulative hierarchy. In other words, the axiom of foundation states that there are no sets in ZF besides the ones that can be constructed bottom-up by the iterative procedure just described. Since in a cumulative hierarchy, there can be no sets containing themselves, no universal set, and no non-wellfounded sets, none of the known paradoxes can immediately be formulated in the theory. This does obviously not in itself ensure the consistency of ZF, but at least it illustrates how the idea of a set hierarchy plays a significant role in ZF as well. ZF has a privileged status among set theories, as it is today the most widely acknowledged candidate for a formal foundation of mathematics.