cumulative hierarchy

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

$W_\alpha \subseteq W_{\alpha + 1}$
If $\lambda$ is a limit ordinal, then $W_\lambda = \bigcup_{\alpha < \lambda} W_{\alpha}$

Some authors additionally require that $W_{\alpha + 1} \subseteq \mathcal P(W_\alpha)$ .{{cn|reason=Give an example citation for each additional requirements.|date=June 2019}}

The union $W = \bigcup_{\alpha \in \mathrm{On}} W_\alpha$ of the sets of a cumulative hierarchy is often used as a model of set theory.{{cn|date=June 2019}}

The phrase "the cumulative hierarchy" usually refers to the von Neumann universe, which has $W_{\alpha + 1} = \mathcal P(W_\alpha)$ .

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 $\mathrm{V}_\alpha$ .
The sets $\mathrm{L}_\alpha$ 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

{{cite book | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition=Third Millennium | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002 }}

Category: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}}