mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, a cumulative hierarchy is a family of sets

W_\alpha

indexed by ordinals

\alpha

such that *

W_\alpha \subseteq W_

* If

\lambda

is a

limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...

, then

W_\lambda = \bigcup_ W_

Some authors additionally require that

W_ \subseteq \mathcal P(W_\alpha)

. The union

W = \bigcup_ 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

von Neumann universe In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory ( ...

, which has

W_ = \mathcal P(W_\alpha)

Reflection principle

A cumulative hierarchy satisfies a form of the

reflection principle In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that, with respect to any given property, resemble the class of all sets. There are several different forms of the reflection principle depending ...

: any

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

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_\alpha

. *The sets

\mathrm_\alpha

of the

constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...

form a cumulative hierarchy. *The

Boolean-valued model In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take v ...

s 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

* Set theory * {{cite journal, last1=Zermelo, first1=Ernst, author1-link=Ernst Zermelo, title=Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, journal=

Fundamenta Mathematicae ''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical sys ...

, volume=16, year=1930, pages=29–47, doi=10.4064/fm-16-1-29-47, url=https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/16/0/92877/uber-grenzzahlen-und-mengenbereiche, doi-access=free