The cumulative hierarchy in set theory is the hierarchy V<sub>α</sub> that is often defined by the transfinite recursion:
- V<sub>0</sub> = emptyset
- V<sub>α+1</sub> = P(V<sub>α</sub>), taking the power set at successor stages
- V<sub>λ</sub> = U{ V<sub>α</sub> | 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 = V<sub>ω</sub> = U V<sub>n</sub>. 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 V<sub>α</sub> = U { P(V<sub>β</sub>) | 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 V<sub>n</sub>, where V<sub>n</sub> = U { P(V<sub>m</sub> | m < n }. This definition needs no base case, and does not split into cases.
One can now prove all the basic facts about the V<sub>α</sub> hierarchy, also without splitting into zero, successor and limit cases. For example, every V<sub>α</sub> is transitive, since by induction it is the union of transitive sets. Similarly, the hierarchy is increasing, etc.