Von Neumann universe
Encyclopedia
In set theory
and related branches of mathematics
, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class
of hereditary
well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory
(ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.
The rank of a well-founded set is defined inductively as the smallest ordinal number
greater than the ranks of all members of the set. In particular, the rank of the empty set
is zero, and every ordinal has a rank equal to itself. The sets in V are divided into a transfinite hierarchy, called the cumulative hierarchy, based on their rank.
indexed by the class of ordinal number
s, in particular, Vα is the set of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α; Vα may be defined by transfinite recursion as follows:
A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines "the set x is in Vα".
The class V is defined to be the union of all the V-stages:
An equivalent definition sets
for each ordinal α, where
is the powerset of X.
The rank of a set S is the smallest α such that
s, then Vω is the set of hereditarily finite set
s, which is a model of set theory without the axiom of infinity
.
Vω+ω is the universe of "ordinary mathematics
", and is a model of Zermelo set theory
. If κ is an inaccessible cardinal
, then Vκ is a model of Zermelo-Fraenkel set theory (ZFC) itself, and Vκ+1 is a model of Morse–Kelley set theory
.
V is not "the set of all sets" for two reasons. First, it is not a set; although each individual stage Vα is a set, their union V is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity) demands that every set is well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (for example is Aczel's anti-foundation axiom
). These non-well-founded set theories are not commonly employed, but are still possible to study.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and related branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
of hereditary
Hereditary set
In set theory, a hereditary set is a set all of whose elements are hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on...
well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
(ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.
The rank of a well-founded set is defined inductively as the smallest ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
greater than the ranks of all members of the set. In particular, the rank of the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
is zero, and every ordinal has a rank equal to itself. The sets in V are divided into a transfinite hierarchy, called the cumulative hierarchy, based on their rank.
Definition
The cumulative hierarchy is a collection of sets Vαindexed by the class of ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s, in particular, Vα is the set of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α; Vα may be defined by transfinite recursion as follows:
- Let V0 be the empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
, {}. - For any ordinal numberOrdinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
β, let Vβ+1 be the power set of Vβ. - For any limit ordinal λ, let Vλ be the unionUnion (set theory)In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of all the V-stages so far: -
A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines "the set x is in Vα".
The class V is defined to be the union of all the V-stages:
An equivalent definition sets
for each ordinal α, where
is the powerset of X.The rank of a set S is the smallest α such that
V and set theory
If ω is the set of natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, then Vω is the set of hereditarily finite set
Hereditarily finite set
In mathematics and set theory, hereditarily finite sets are defined recursively as finite sets consisting of 0 or more hereditarily finite sets.-Formal definition:...
s, which is a model of set theory without the axiom of infinity
Axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
.
Vω+ω is the universe of "ordinary mathematics
Ordinary mathematics
In the philosophy of mathematics, ordinary mathematics is an inexact term, used to distinguish the body of most mathematical work from that of, for example, constructivist, intuitionist, or finitist mathematics....
", and is a model of Zermelo set theory
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
. If κ is an inaccessible cardinal
Inaccessible cardinal
In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable...
, then Vκ is a model of Zermelo-Fraenkel set theory (ZFC) itself, and Vκ+1 is a model of Morse–Kelley set theory
Morse–Kelley set theory
In the foundation of mathematics, Morse–Kelley set theory or Kelley–Morse set theory is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory...
.
V is not "the set of all sets" for two reasons. First, it is not a set; although each individual stage Vα is a set, their union V is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity) demands that every set is well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (for example is Aczel's anti-foundation axiom
Aczel's anti-foundation axiom
Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel in . It states that every accessible pointed directed graph corresponds to a unique set. In particular, the graph consisting of a single vertex with a loop corresponds to a set which contains only itself as element, i.e...
). These non-well-founded set theories are not commonly employed, but are still possible to study.
Philosophical perspectives
There are two approaches to understanding the relationship of the von Neumann universe V to ZFC (along with many variations of each approach, and shadings between them). Roughly, formalists will tend to view V as something that flows from the ZFC axioms (for example, ZFC proves that every set is in V). On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language. A possible middle position is that the mental picture of the von Neumann hierarchy provides the ZFC axioms with a motivation (so that they are not arbitrary), but does not necessarily describe objects with real existence.See also
- Universe (mathematics)Universe (mathematics)In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...
- Constructible universeConstructible universeIn mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
- Grothendieck universeGrothendieck universeIn mathematics, a Grothendieck universe is a set U with the following properties:# If x is an element of U and if y is an element of x, then y is also an element of U...
- Inaccessible cardinalInaccessible cardinalIn set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable...
- S (Boolos 1989)
- John von NeumannJohn von NeumannJohn von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...