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$The cumulative hierarchy is a collection of sets V? indexed by the class of ordinal numbers, in particular, V? is the set of all sets having ranks less than ?. Thus there is one set V? for each ordinal number ?; V? may be defined by transfinite recursion as follows: Let V0 be the empty set, {}: V_0 := \{\} . For any ordinal number ?, let V?+1 be the power set of V?: V_{\beta+1} := \mathcal{P} (V_\beta) . For any limit ordinal ?, let V? be the union of all the V-stages so far: V_\lambda := \bigcup_{\beta < \lambda} V_\beta . A crucial fact about this definition is that there is a single formula ?(?,x) in the language of ZFC that defines "the set x is in V?". The class V is defined to be the union of all the V-stages: V := \bigcup_{\alpha} V_\alpha. An equivalent definition sets V_\alpha := \bigcup_{\beta < \alpha} \mathcal{P} (V_\beta) for each ordinal ?, where \mathcal{P} (X) \! is the powerset of X. The rank of a set S is the smallest ? such that S \subseteq V_\alpha \,.$
The cumulative hierarchy is a…