In mathematics, specifically set theory, a cumulative hierarchy is a family of sets indexed by ordinals such that
Some authors additionally require that or that .[ citation needed]
The union of the sets of a cumulative hierarchy is often used as a model of set theory.[ citation needed]
The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy of the von Neumann universe with introduced by Zermelo (1930).
A cumulative hierarchy satisfies a form of the reflection principle: any formula in the language of set theory that holds in the union of the hierarchy also holds in some stages .