Cumulative hierarchy

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 W₀ 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.