1. Reflection principle

2. Examples

3. References

In mathematical set theory, a cumulative hierarchy is a family of sets W_α indexed by ordinals α such that

• W_α⊆W_α+1
• If α is a limit then W_α = ∪_β<α W_β

It is also sometimes assumed that W_α+1⊆P(W_α) 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 V_α of the Von Neumann universe with V_α+1=P(V_α) introduced by {{harvtxt|Zermelo|1930}}

Reflection principle

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 W_α.

Examples

• The Von Neumann universe is built from a cumulative hierarchy V_α.
• The sets L_α 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

• {{cite book | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition=Third Millennium | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002 }}

• {{cite journal|ref=harv|last1=Zermelo|first1=Ernst|author1-link=Ernst Zermelo|title=Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre|journal=Fundamenta Mathematicae|volume=16|year=1930|pages=29–47}}

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