In
mathematics, specifically
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
, a cumulative hierarchy is a family of
sets indexed by
ordinals such that
*
* If
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
Some authors additionally require that
or that
.
The
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
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
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 (ZF ...
with
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
of the hierarchy also holds in some stages
.
Examples
* The von Neumann universe is built from a cumulative hierarchy
.
*The sets
of the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
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
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the ax ...
) 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