cumulative hierarchy (original) (raw)

Every set is a subset of Vα for some ordinal α, and the least such α is called the rank of the set. It can be shown that the rank of an ordinal is itself, and in general the rank of a set Xis the least ordinal greater than the rank of every element of X. For each ordinal α, the set Vα is the set of all sets of rank less than α, and Vα+1∖Vα is the set of all sets of rank α.

Note that the previous paragraph makes use of the Axiom of Foundation: if this axiom fails, then there are sets that are not subsets of any Vαand therefore have no rank. The previous paragraph also assumes that we are using a set theory such as ZF, in which elements of sets are themselves sets.

Each Vα is a transitive set. Note that V0=0, V1=1 and V2=2, but for α>2 the set Vα is never an ordinal, because it has the element {1}, which is not an ordinal.