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In set theory
Set theory

Set theory is the branch of mathematics that studies Set , 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, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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".

Every set is a class, no matter which foundation is chosen.






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In set theory
Set theory

Set theory is the branch of mathematics that studies Set , 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, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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".

Every set is a class, no matter which foundation is chosen. A class that is not a set is (informally in Zermelo–Fraenkel) 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-order. They are usually identified with hereditarily transitive sets....
s, and the class of all sets, are proper classes in many formal systems.

Various important concepts in mathematics are commonly described with classes. Examples include large categories
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
 and the class-field of surreal number
Surreal number

In mathematics, the surreal number system is an continuum containing the real number as well as infinite and infinitesimal, respectively larger or smaller in absolute value than any positive real number....
s.

In ZF set theory, classes exist only in the metalanguage
Metalanguage

In logic and linguistics, a metalanguage is a language used to make statements about statements in another language which is called the object language....
, as equivalence classes of logical formulas. The axioms of ZF do not apply to classes. However, if an inaccessible cardinal
Inaccessible cardinal

In set theory, an uncountable set regular cardinal is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal....
 ? 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; 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. In other, less standard 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....
 or the theory of semiset
Semiset

In set theory, a semiset is a class which is contained in a Set .The theory of semisets was proposed and developed by Czech Republic mathematicians Petr Vopenka and Petr H?jek ....
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.

The paradoxes of naive set theory
Naive set theory

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most contemporary mathematics....
 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 paradox
Russell's paradox

Part of fundamental mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Gottlob Frege 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. One way to prove that a class is proper is to place it in bijection with the class of ordinals; see, for instance, the proof that there is no free complete lattice
Free lattice

In mathematics, in the area of order theory, a free lattice is the free object corresponding to a Lattice . As free objects, they have the universal property....
.

The word "class" was sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.