In the
foundations of mathematics
Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
, von Neumann–Bernays–Gödel set theory (NBG) is an
axiomatic 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 mathematics, ...
that is a
conservative extension In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superthe ...
of
Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the
notion
Notion or Notions may refer to:
Software
* Notion (music software), a music composition and performance program
* Notion (productivity software), a note-taking and project-management program from Notion Labs Inc.
* Notion (window manager), the s ...
of
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 ...
, which is a collection of
sets defined by a
formula whose
quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all
ordinals.
Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.
A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of
atomic formula
In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...
s (
membership
Member may refer to:
* Military jury, referred to as "Members" in military jargon
* Element (mathematics), an object that belongs to a mathematical set
* In object-oriented programming, a member of a class
** Field (computer science), entries in ...
and
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
) and finitely many
logical symbols, only finitely many
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the
set-theoretic paradoxes, and for stating the
axiom of global choice In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an ele ...
, which is stronger than ZFC's
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.
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 ...
introduced classes into set theory in 1925. The
primitive notion
In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an ...
s of his theory were
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
and
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
. Using these notions, he defined class and set.
[; English translation: .] Paul Bernays
Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of ...
reformulated von Neumann's theory by taking class and set as primitive notions.
[, pp. 66–67.] Kurt Gödel simplified Bernays' theory for his
relative consistency proof of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
and the
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
.
[.]
Classes in set theory
The uses of classes
Classes have several uses in NBG:
* They produce a finite axiomatization of set theory.
* They are used to state a "very strong form of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
"
[.]—namely, the
axiom of global choice In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an ele ...
: There exists a global choice function
defined on the class of all nonempty sets such that
for every nonempty set
This is stronger than ZFC's axiom of choice: For every set
of nonempty sets, there exists a
choice function
A choice function (selector, selection) is a mathematical function ''f'' that is defined on some collection ''X'' of nonempty sets and assigns some element of each set ''S'' in that collection to ''S'' by ''f''(''S''); ''f''(''S'') maps ''S'' to ...
defined on
such that
for all
* The
set-theoretic paradoxes are handled by recognizing that some classes cannot be sets. For example, assume that the class
of all
ordinals is a set. Then
is a
transitive set
well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-orde ...
ed by
. So, by definition,
is an ordinal. Hence,
, which contradicts
being a well-ordering of
Therefore,
is not a set. Because a class that is not a set is called 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 ...
,
is a proper class.
* Proper classes are useful in constructions. In his proof of the relative consistency of the axiom of global choice and the
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
, Gödel used proper classes to build 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 ...
. He constructed a function on the class of all ordinals that, for each ordinal, builds a constructible set by applying a set-building operation to previously constructed sets. The constructible universe is the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of this function.
Axiom schema versus class existence theorem
Once classes are added to the language of ZFC, it is easy to transform ZFC into a set theory with classes. First, the
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
Formal definition
An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...
of class comprehension is added. This axiom schema states: For every formula
that quantifies only over sets, there exists a class
consisting of the satisfying the formula—that is,
Then the
axiom schema 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 ...
is replaced by a
single axiom that uses a class. Finally, ZFC's
axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements ...
is modified to handle classes: If two classes have the same elements, then they are identical. The other axioms of ZFC are not modified.
This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced.
To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many
class existence axioms. Then these axioms are used to prove the class existence theorem, which implies every instance of the axiom schema.
The
proof of this theorem requires only seven class existence axioms, which are used to convert the construction of a formula into the construction of a class satisfying the formula.
Axiomatization of NBG
Classes and sets
NBG has two types of objects: classes and sets. Intuitively, every set is also a class. There are two ways to axiomatize this. Bernays used
many-sorted logic
Many-sorted logic can reflect formally our intention not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming. Both functional and assertive " parts of speech ...
with two sorts: classes and sets.
Gödel avoided sorts by introducing primitive predicates:
for "
is a class" and
for "
is a set" (in German, "set" is ''Menge''). He also introduced axioms stating that every set is a class and that if class
is a member of a class, then
is a set.
[.] Using predicates is the standard way to eliminate sorts.
Elliott Mendelson
Elliott Mendelson (May 24, 1931 – May 7, 2020) was an American logician. He was a professor of mathematics at Queens College of the City University of New York, and the Graduate Center, CUNY. He was Jr. Fellow, Society of Fellows, Harvard Univ ...
modified Gödel's approach by having everything be a class and defining the set predicate
as
This modification eliminates Gödel's class predicate and his two axioms.
Bernays' two-sorted approach may appear more natural at first, but it creates a more complex theory. In Bernays' theory, every set has two representations: one as a set and the other as a class. Also, there are two
membership relations: the first, denoted by "∈", is between two sets; the second, denoted by "η", is between a set and a class.
This redundancy is required by many-sorted logic because variables of different sorts range over disjoint subdomains of the
domain of discourse.
The differences between these two approaches do not affect what can be proved, but they do affect how statements are written. In Gödel's approach,
where
and
are classes is a valid statement. In Bernays' approach this statement has no meaning. However, if
is a set, there is an equivalent statement: Define "set
represents class
" if they have the same sets as members—that is,
The statement
where set
represents class
is equivalent to Gödel's
The approach adopted in this article is that of Gödel with Mendelson's modification. This means that NBG is an
axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
in
first-order predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
with
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
, and its only
primitive notion
In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an ...
s are class and the membership relation.
Definitions and axioms of extensionality and pairing
A set is a class that belongs to at least one class:
is a set if and only if
.
A class that is not a set is called a proper class:
is a proper class if and only if
.
Therefore, every class is either a set or a proper class, and no class is both.
Gödel introduced the convention that uppercase variables range over classes, while lowercase variables range over sets.
Gödel also used names that begin with an uppercase letter to denote particular classes, including functions and
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
s defined on the class of all sets. Gödel's convention is used in this article. It allows us to write:
*
instead of
*
instead of
The following axioms and definitions are needed for the proof of the class existence theorem.
Axiom of extensionality. If two classes have the same elements, then they are identical.
: