Cumulative hierarchy

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 . The union 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 of the von Neumann universe with introduced by . 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 . 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 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.