Cumulative hierarchy

In mathematics, specifically set theory, a cumulative hierarchy is a family of sets Wα indexed by ordinals α such that

  • Wα ⊆ Wα+1
  • If α is a limit ordinal, then Wα = ∪β<α Wβ

Some authors additionally require 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.

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 in the language of set theory that holds in the union W of the hierarchy also holds in some stages Wα.

Examples
  • 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.

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.CS1 maint: ref=harv (link)
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