In mathematics, specifically set theory, a cumulative hierarchy is a family of sets indexed by ordinals such that
 If is a limit ordinal, then
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).
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Transcription
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 of the hierarchy also holds in some stages .
Examples
 The von Neumann universe is built from a cumulative hierarchy .
 The sets of the constructible universe form a cumulative hierarchy.
 The Booleanvalued 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: SpringerVerlag. ISBN 9783540440857. 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/fm1612947.