The cumulative hierarchy in set theory is the hierarchy V_α that is often defined by the transfinite recursion:

• V₀ = emptyset
• V_α+1 = P(V_α), taking the power set at successor stages
• V_λ = U{ V_α | 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_ω = U V_n. 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_α = U { P(V_β) | 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_n, where V_n = U { P(V_m | m < n }. This definition needs no base case, and does not split into cases.

The cumulative hierarchy in set theory is the hierarchy V_α that is often defined by the transfinite recursion:

• V₀ = emptyset
• V_α+1 = P(V_α), taking the power set at successor stages
• V_λ = U{ V_α | 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_ω = U V_n. 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_α = U { P(V_β) | 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_n, where V_n = U { P(V_m | 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_α hierarchy, also without splitting into zero, successor and limit cases. For example, every V_α is transitive, since by induction it is the union of transitive sets. Similarly, the hierarchy is increasing, etc.