In
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 ...
and related branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differentl ...
of
hereditary
Heredity, also called inheritance or biological inheritance, is the passing on of traits from parents to their offspring; either through asexual reproduction or sexual reproduction, the offspring cells or organisms acquire the genetic inform ...
well-founded set
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s&n ...
s. This collection, which is formalized by
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
, although it was first published by
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
in 1930.
The rank of a well-founded set is defined inductively as the smallest
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
greater than the ranks of all members of the set. In particular, the rank of the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is zero, and every ordinal has a rank equal to itself. The sets in ''V'' are divided into the
transfinite Transfinite may refer to:
* Transfinite number, a number larger than all finite numbers, yet not absolutely infinite
* Transfinite induction, an extension of mathematical induction to well-ordered sets
** Transfinite recursion
* Transfinite arithmet ...
hierarchy ''V
α'', called the cumulative hierarchy, based on their rank.
Definition
The cumulative hierarchy is a collection of sets ''V''
α
indexed by the class of
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s; in particular, ''V''
α is the set of all sets having ranks less than α. Thus there is one set ''V''
α for each ordinal number α. ''V''
α may be defined by
transfinite recursion
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...
as follows:
* Let ''V''
0 be the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
:
*:
* For any
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
β, let ''V''
β+1 be the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of ''V''
β:
*:
* For any
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 an ...
λ, let ''V''
λ be 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 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 states "the set ''x'' is in ''V''
α".
The sets ''V''
α are called stages or ranks.
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
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postu ...
of
.
The rank of a set ''S'' is the smallest α such that
Another way to calculate rank is:
:
.
Finite and low cardinality stages of the hierarchy
The first five von Neumann stages ''V''
0 to ''V''
4 may be visualized as follows. (An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.)
![Von Neumann universe 4](https://upload.wikimedia.org/wikipedia/commons/8/83/Von_Neumann_universe_4.png)
This sequence exhibits
tetrational
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
growth. The set ''V''
5 contains 2
16 = 65536 elements; the set ''V''
6 contains 2
65536 elements, which very substantially exceeds the
number of atoms in the known universe; and for any natural ''n'', the set ''V''
''n+1'' contains 2 ↑↑ ''n'' elements using
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...
. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set ''V''
ω has the same cardinality as ω. The set ''V''
ω+1 has the same cardinality as the set of real numbers.
Applications and interpretations
Applications of ''V'' as models for set theories
If ω is the set of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s, then ''V''
ω is the set of
hereditarily finite set
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to t ...
s, which is a
model
A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
of set theory without the
axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...
.
''V''
ω+ω is the
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
of "ordinary mathematics", and is a model of
Zermelo set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It be ...
. A simple argument in favour of the adequacy of ''V''
ω+ω is the observation that ''V''
ω+1 is adequate for the integers, while ''V''
ω+2 is adequate for the real numbers, and most other normal mathematics can be built as relations of various kinds from these sets without needing the
axiom of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
to go outside ''V''
ω+ω.
If κ is an
inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
, then ''V''
κ is a model of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(ZFC) itself, and ''V''
κ+1 is a model of
Morse–Kelley set theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
. (Note that every ZFC model is also a ZF model, and every ZF model is also a Z model.)
Interpretation of ''V'' as the "set of all sets"
V is not "the
set of all sets
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
" for two reasons. First, it is not a set; although each individual stage ''V''
α is a set, their union ''V'' is a
proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
. Second, the sets in ''V'' are only the well-founded sets. 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 ...
(or regularity) demands that every set be 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 (an example is
Aczel's anti-foundation axiom
In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by , as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to exact ...
). These non-well-founded set theories are not commonly employed, but are still possible to study.
A third objection to the "set of all sets" interpretation is that not all sets are necessarily "pure sets", which are constructed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion of
urelement
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual.
Theory
There ...
s, from which he constructed a transfinite recursive hierarchy in 1930.
Such urelements are used extensively in
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, particularly in Fraenkel-Mostowski models.
[.]
''V'' and the axiom of regularity
The formula ''V'' = ⋃
α''V''
α is often considered to be a theorem, not a definition.
Roitman states (without references) that the realization that the
axiom of regularity
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 axi ...
is equivalent to the equality of the universe of ZF sets to the cumulative hierarchy is due to von Neumann.
[. See page 79.]
The existential status of ''V''
Since the class ''V'' may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research i ...
, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.
The integrity of the von Neumann universe depends fundamentally on the integrity of the
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s, which act as the rank parameter in the construction, and the integrity of
transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...
, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers. The integrity of the construction of ''V'' by transfinite induction may be said to have then been established in Zermelo's 1930 paper.
History
The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Moore (1982) to be inaccurately attributed to
von Neumann Von Neumann may refer to:
* John von Neumann (1903–1957), a Hungarian American mathematician
* Von Neumann family
* Von Neumann (surname), a German surname
* Von Neumann (crater), a lunar impact crater
See also
* Von Neumann algebra
* Von Ne ...
.
[. See page 279 for the assertion of the false attribution to von Neumann. See pages 270 and 281 for the attribution to Zermelo.] The first publication of the von Neumann universe was by
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
in 1930.
[. See particularly pages 36–40.]
Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory and Neumann's own set theory (which later developed into
NBG set theory NBG may refer to:
*Namibian Black German, a pidgin variant of the German language native to Namibia
*Natal Border Guard, a military unit of the Colony of Natal during the Anglo-Zulu War
*National Bank of Georgia
*National Bank of Greece
*Naval Air ...
). In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. The presentations of the von Neumann universe by Bernays
[. See pages 203–209.] and Mendelson
[. See page 202.] both give credit to von Neumann for the transfinite induction construction method, although not for its application to the construction of the universe of ordinary sets.
The notation ''V'' is not a tribute to the name of von Neumann. It was used for the universe of sets in 1889 by Peano, the letter ''V'' signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals.
[. See pages VIII and XI.] Peano's notation ''V'' was adopted also by Whitehead and Russell for the class of all sets in 1910.
[. See page 229.] The ''V'' notation (for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfinite induction. Paul Cohen
[. See page 88.] explicitly attributes his use of the letter ''V'' (for the class of all sets) to a 1940 paper by Gödel,
[.] although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.
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)
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation.
In set theory, universes are ...
*
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 ...
*
Grothendieck universe In 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''. (''U'' is a transitive set.)
# If ''x'' and ''y'' a ...
*
Inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
*
S (set theory)
S is an axiomatic set theory set out by George Boolos in his 1989 article, "Iteration Again". S, a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. Boolos designed S to embody his understanding of the ...
Notes
References
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* . English translation:
*
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{{Mathematical logic
John von Neumann
Set-theoretic universes