Von Neumann Universe - Definition

Definition

The cumulative hierarchy is a collection of sets V_α indexed by the class of ordinal numbers, in particular, V_α is the set of all sets having ranks less than α. Thus there is one set V_α for each ordinal number α; V_α may be defined by transfinite recursion as follows:

• Let V₀ be the empty set, {}:

• For any ordinal number β, let V_β+1 be the power set of V_β:

• For any limit ordinal λ, let V_λ 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 V_α".

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