The cumulative hierarchy in set theory is the hierarchy Vα that is often defined by the transfinite recursion:
- V0 = emptyset
- Vα+1 = P(Vα), taking the power set at successor stages
- Vλ = U{ Vα | α<λ }, taking unions at limits.
And if one wants to consider only the class HF of hereditarily finite sets, one restricts to the natural number induction α=n, leading to HF = Vω = U Vn. These definitions, however, split into separate cases for zero, successor and limit.
An equivalent definition, however, completely avoids this split. Namely:
- For any ordinal α, let Vα = U { P(Vβ) | β < α }.
This is easily seen to be equivalent to the previous definition.
Similarly, one can define the hereditary finite sets HF as the union of Vn, where Vn = U { P(Vm | m < n }. This definition needs no base case, and does not split into cases.
One can now prove all the basic facts about the Vα hierarchy, also without splitting into zero, successor and limit cases. For example, every Vα is transitive, since by induction it is the union of transitive sets. Similarly, the hierarchy is increasing, etc.