Cumulative hierarchy
In mathematics, specifically set theory, a cumulative hierarchy is a family of sets [math]\displaystyle{ W_\alpha }[/math] indexed by ordinals [math]\displaystyle{ \alpha }[/math] such that
- [math]\displaystyle{ W_\alpha \subseteq W_{\alpha + 1} }[/math]
- If [math]\displaystyle{ \lambda }[/math] is a limit ordinal, then [math]\displaystyle{ W_\lambda = \bigcup_{\alpha \lt \lambda} W_{\alpha} }[/math]
Some authors additionally require that [math]\displaystyle{ W_{\alpha + 1} \subseteq \mathcal P(W_\alpha) }[/math] or that [math]\displaystyle{ W_0 \ne \emptyset }[/math].[citation needed]
The union [math]\displaystyle{ W = \bigcup_{\alpha \in \mathrm{On}} W_\alpha }[/math] 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 [math]\displaystyle{ \mathrm{V}_\alpha }[/math] of the von Neumann universe with [math]\displaystyle{ \mathrm{V}_{\alpha + 1} = \mathcal P(W_\alpha) }[/math] 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 [math]\displaystyle{ W }[/math] of the hierarchy also holds in some stages [math]\displaystyle{ W_\alpha }[/math].
Examples
- The von Neumann universe is built from a cumulative hierarchy [math]\displaystyle{ \mathrm{V}_\alpha }[/math].
- The sets [math]\displaystyle{ \mathrm{L}_\alpha }[/math] 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.
- 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. https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/16/0/92877/uber-grenzzahlen-und-mengenbereiche.
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