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 $$W_{\alpha + 1} \subseteq \mathcal P(W_\alpha)$$ or that $$W_0 \ne \emptyset$$.

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.

The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy $$\mathrm{V}_\alpha$$ of the von Neumann universe with $$\mathrm{V}_{\alpha + 1} = \mathcal P(W_\alpha)$$ introduced by.

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.