Definition
The cumulative hierarchy is a collection of sets Vα indexed by the class of ordinal numbers, in particular, Vα is the set of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α; Vα may be defined by transfinite recursion as follows:
- Let V0 be the empty set, {}:
- For any ordinal number β, let Vβ+1 be the power set of Vβ:
- For any limit ordinal λ, let Vλ be the union of all the V-stages so far:
A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines "the set x is in Vα".
The class V is defined to be the union of all the V-stages:
An equivalent definition sets
for each ordinal α, where is the powerset of .
The rank of a set S is the smallest α such that
Read more about this topic: Von Neumann Universe
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