Cumulative hierarchy

In mathematics, specifically set theory, a cumulative hierarchy is a family of sets $W_\alpha$ indexed by ordinals $\alpha$ such that

* $W_\alpha \subseteq W_$
* If $\lambda$ is a limit ordinal, then $W_\lambda = \bigcup_ W_$

Some authors additionally require that $W_ \subseteq \mathcal P(W_\alpha)$ or that $W_0 \ne \emptyset$.

The union $W = \bigcup_ W_\alpha$ 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 $\mathrm_\alpha$ of the von Neumann universe with $\mathrm_ = \mathcal P(W_\alpha)$ introduced by .

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 $W_\alpha$.

Examples

* The von Neumann universe is built from a cumulative hierarchy $\mathrm_\alpha$.
*The sets $\mathrm_\alpha$ 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

*
Set theory
* {{cite journal, last1=Zermelo, first1=Ernst, author1-link=Ernst Zermelo, title=Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, journal= Fundamenta Mathematicae, volume=16, year=1930, pages=29–47, doi=10.4064/fm-16-1-29-47, url=https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/16/0/92877/uber-grenzzahlen-und-mengenbereiche, doi-access=free