# Cumulative hierarchy

In mathematical set theory, a cumulative hierarchy is a family of sets Wα indexed by ordinals α such that

• WαWα+1
• If α is a limit then Wα = ∪β<α Wβ

It is also sometimes assumed that Wα+1P(Wα) or that W0 is empty.

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) introduced by Zermelo (1930)

## Reflection principleEdit

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.

## ExamplesEdit

• 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.