Cumulative hierarchy

In mathematics, specifically set theory, a cumulative hierarchy is a family of sets $W_{α}$ indexed by ordinal numbers $α$ such that

$W_{α} \subseteq W_{α + 1}$ , and
if $λ$ is a limit ordinal, then $W_{λ} = ⋃_{α < λ} W_{α}$ .

Some authors additionally require that $W_{α + 1} \subseteq 𝒫 (W_{α})$ .

The union over all ordinals $W = ⋃_{α \in O n} W_{α}$ 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 von Neumann hierarchy, which has $W_{α + 1} = 𝒫 (W_{α})$ .

Reflection principle

Cumulative hierarchies satisfy 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 stage $W_{α}$ .

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) [1978]. Set Theory. Springer Monographs in Mathematics (3rd millennium ed.). Berlin, Heidelberg: Springer. ISBN 978-3-540-44085-7. Zbl 1007.03002.

Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche". Fundamenta Mathematicae (in German). 16: 29–47. doi:10.4064/fm-16-1-29-47. JFM 56.0082.02.

This set theory–related article is a stub. You can help Wikipedia by adding missing information.