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Definition

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 α. Thus there is one set V for each ordinal number α. V may be defined by transfinite recursion as follows:

A crucial fact about this definition is that there is a single formula φ(α, x) in the language of ZFC that states "the set x is in V ".

The sets V are called stages or ranks.

The class V is defined to be the union of all the V -stages: