Cumulative hierarchy
Appearance
In mathematics, specifically set theory, a cumulative hierarchy is a family of sets indexed by ordinals such that
- If is a limit ordinal, then
Some authors additionally require that or that .[citation needed]
The union 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 of the von Neumann universe with 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 of the hierarchy also holds in some stages .
Examples
- The von Neumann universe is built from a cumulative hierarchy .
- The sets of the constructible universe form a cumulative hierarchy.
- The Boolean-valued models constructed by forcing are built using a cumulative hierarchy.
- The axiom of foundation) form a cumulative hierarchy whose union satisfies the axiom of foundation.
References
- Zbl 1007.03002.
- .