Cumulative hierarchy
In mathematical set theory, a cumulative hierarchy is a family of sets Wα indexed by ordinals α such that
It is also sometimes assumed that Wα+1⊆P(Wα) or that W0 is empty.
The union W 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 Vα of the Von Neumann universe with Vα+1=P(Vα) introduced by Zermelo (1930)
A cumulative hierarchy satisfies a form of the reflection principle: any formula of the language of set theory that holds in the union W of the hierarchy also holds in some stages Wα.
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