Class (set theory)
Encyclopedia
In set theory
and its applications throughout mathematics
, a class is a collection of sets (or sometimes other mathematical objects) which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF set theory
, the notion of class is informal, whereas other set theories, such as NBG set theory
, axiomatize the notion of "class", e.g., as entities that are not members of another entity.
Every set is a class, no matter which foundation is chosen. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal number
s, and the class of all sets, are proper classes in many formal systems.
Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.
s, the class of all vector space
s, and many others. In category theory
, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.
The surreal number
s are a proper class of objects that has the properties of a field
.
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers.
One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice.
suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox
suggests that the class of all ordinal numbers is proper.
, as equivalence classes of logical formulas. For example, if
is a structure
interpreting ZF, then the metalanguage expression
is interpreted in 
by the collection of all the elements from the domain of 
; that is, all the sets in 
. So we can identify the "class of all sets" with the predicate x=x or any equivalent predicate.
Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal
κ is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe
), and its subsets can be thought of as "classes".
Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the set existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a conservative extension
of ZF.
Morse–Kelley set theory
admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its set existence axioms. This causes MK to be strictly stronger than both NBG and ZF.
In other set theories, such as New Foundations
or the theory of semiset
s, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.
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 its applications throughout 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...
, a class is a collection of sets (or sometimes other mathematical objects) which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF 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...
, the notion of class is informal, whereas other set theories, such as NBG set theory
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
, axiomatize the notion of "class", e.g., as entities that are not members of another entity.
Every set is a class, no matter which foundation is chosen. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all 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, and the class of all sets, are proper classes in many formal systems.
Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.
Examples
The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s, the class of all vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, and many others. In category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.
The surreal number
Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number...
s are a proper class of objects that has the properties of a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
.
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers.
One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice.
Paradoxes
The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper. For example, Russell's paradoxRussell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
suggests that the class of all ordinal numbers is proper.
Classes in formal set theories
ZF set theory does not formalize the notion of classes. They can instead be described in the metalanguageMetalanguage
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined. In logic and linguistics, a metalanguage is a language used to make statements about statements in another language...
, as equivalence classes of logical formulas. For example, if
is a structureStructure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
interpreting ZF, then the metalanguage expression
is interpreted in by the collection of all the elements from the domain of ; that is, all the sets in . So we can identify the "class of all sets" with the predicate x=x or any equivalent predicate.Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if 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...
κ is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe
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...
), and its subsets can be thought of as "classes".
Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the set existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a conservative extension
Conservative extension
In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...
of ZF.
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...
admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its set existence axioms. This causes MK to be strictly stronger than both NBG and ZF.
In other set theories, such as New Foundations
New Foundations
In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
or the theory of semiset
Semiset
In set theory, a semiset is a proper class which is contained in a set.The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek . It is based on a modification of the von Neumann-Bernays-Gödel set theory; in standard NBG, the existence of semisets is...
s, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.