In mathematical set theory, a cumulative hierarchy is a family of sets Wα indexed by ordinals α such that
- If α is a limit or 0 then Wα = ∪β<α Wβ
It is also sometimes assumed that Wα+1⊆P(Wα)
The union W of the sets of a cumulative hierarchy is often used as a model of set theory.
The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy Vα of the Von Neumann universe with Vα+1=P(Vα).
A cumulative hierarchy satisfies a form of the reflection principle: any formula of the language of set theory that holds in the union W of the hierarchy also holds in some stages Wα.
- The Von Neumann universe is built from a cumulative hierarchy Vα.
- The sets Lα of the constructible universe form a cumulative hierarchy.
- The Boolean valued models constructed by forcing are built using a cumulative hierarchy.
- The well founded sets in a model of set theory (possibly not satisfying the axiom of foundation) form a cumulative hierarchy whose union satisfies the axiom of foundation.