# 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 principle

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.

## 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.