They are constructed by the following rules:
The set of all hereditarily finite sets is denoted Vω. If we denote P(S) for the power set of S, Vω can also be constructed by first taking the empty set written V0, then V1 = P(V0), V2 = P(V1),..., Vk = P(Vk−1),... Then
The hereditarily finite sets are a subclass of the Von Neumann universe. They are a model of the axioms consisting of the axioms of set theory with the axiom of infinity replaced by its negation, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.
Equivalently, a set is hereditarily finite if and only if its transitive closure is finite. Vω is also symbolized by , meaning hereditarily of cardinality less than . See also hereditarily countable set.