In mathematics, specifically set theory, a **cumulative hierarchy** is a family of sets *W*_{α} indexed by ordinals α such that

*W*_{α}⊆*W*_{α+1}- If α is a limit ordinal, then
*W*_{α}= ∪_{β<α}*W*_{β}

Some authors additionally require that *W*_{α+1} ⊆ *P*(*W*_{α}) or that *W*_{0} is empty.^{[citation needed]}

The union *W* 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 *V*_{α} of the von Neumann universe with *V*_{α+1} = *P*(*V*_{α}) introduced by Zermelo (1930).

## Reflection principle

A cumulative hierarchy satisfies a form of the reflection principle: any formula in the language of set theory that holds in the union *W* of the hierarchy also holds in some stages *W*_{α}.

## Examples

- The von Neumann universe is built from a cumulative hierarchy
*V*_{α}. - The sets
*L*_{α}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.

## References

- Jech, Thomas (2003).
*Set Theory*. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002. - Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre".
*Fundamenta Mathematicae*.**16**: 29–47. doi:10.4064/fm-16-1-29-47.