- If α is a limit then Wα = ∪β<α Wβ
It is also sometimes assumed that Wα+1⊆P(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.
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.
- 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.