In

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, ...

and its applications throughout 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 ...

, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to Consumables, consume, alte ...

that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid Russell's paradox
In mathematical logic
Mathematical logic is the study of formal logic within 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 ...

(see ). The precise definition of "class" depends on foundational context. In work on 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 suc ...

, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory, Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the Primi ...

, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.
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
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, and the class of all sets, are proper classes in many formal systems.
In Quine's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong.
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 rather perhaps take place without considering that certain classes can fail to be sets.
Examples

The collection of allalgebraic structure
In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...

s of a given type will usually be a proper class. Examples include the class of all group
A group is a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...

s, the class of all vector space
In 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 ...

s, and many others. In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...

, a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
*Categories (Aristotle), ''Categories'' (Aristotle)
*Category (Kant)
...

whose collection of objects forms a proper class (or whose collection of morphism
In 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 mod ...

s forms a proper class) is called a large category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows Assoc ...

.
The surreal number
In mathematics, the surreal number system is a total order, totally ordered proper class containing the real numbers as well as Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value than any ...

s are a proper class of objects that have 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 number
In 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 ...

s.
One way to prove that a class is proper is to place it in bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procure ...

complete lattice
In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...

on three or more generators.
Paradoxes

The paradoxes of naive set theory can be explained in terms of the inconsistenttacit assumption
A tacit assumption or implicit assumption is an assumption that underlies a logical argument, course of action, Decision-making, decision, or judgment that is not explicitly voiced nor necessarily understood by the decision maker or judge. These as ...

that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox
In mathematical logic
Mathematical logic is the study of formal logic within 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 ...

suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox
In 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 ...

suggests that the class of all ordinal numbers
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 ...

is proper. The paradoxes do not arise with classes because there is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a Russell paradox for classes. A conglomerate, on the other hand, can have proper classes as members, although the ''theory'' of conglomerates is not yet well-established.
Classes in formal set theories

ZF set theory ZF, Z-F, or Zf may refer to:
Businesses and organizations
* ZF Friedrichshafen
ZF Friedrichshafen AG, also known as ZF Group, originally ''Zahnradfabrik Friedrichshafen'', and commonly abbreviated to ZF (ZF = "Zahnradfabrik" = "Cogwheel Factor ...

does not formalize the notion of classes, so each formula with classes must be reduced syntactically to a formula without classes. For example, one can reduce the formula $A\; =\; \backslash $ to $\backslash forall\; x(x\; \backslash in\; A\; \backslash leftrightarrow\; x=x)$. Semantically, in a metalanguage
In logic and linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly it ...

, the classes can be described as equivalence classes
In 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 ...

of logical formulas: If $\backslash mathcal\; A$ is a structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...

interpreting ZF, then the object language "class-builder expression" $\backslash $ is interpreted in $\backslash mathcal\; A$ by the collection of all the elements from the domain of $\backslash mathcal\; A$ on which $\backslash lambda\; x\backslash phi$ holds; thus, the class can be described as the set of all predicates equivalent to $\backslash phi$ (which includes $\backslash phi$ itself). In particular, one can identify the "class of all sets" with the set of all predicates equivalent to $x\; =\; x.$
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
In set theory, an uncountable set, uncountable cardinal number, 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 ...

$\backslash kappa$ is assumed, then the sets of smaller rank form a model of ZF (a 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 ...

), and its subsets can be thought of as "classes".
In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula $\backslash Phi(x,y)$ with the property that for any set $x$ there is no more than one set $y$ such that the pair $(x,y)$ satisfies $\backslash Phi.$ For example, the class function mapping each set to its successor may be expressed as the formula $y\; =\; x\; \backslash cup\; \backslash .$ The fact that the ordered pair $(x,y)$ satisfies $\backslash Phi$ may be expressed with the shorthand notation $\backslash Phi(x)\; =\; y.$
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 class 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
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 philoso ...

admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its class existence axioms. This causes MK to be strictly stronger than both NBG and ZF.
In other set theories, such as New Foundations
In mathematical logic, New Foundations (NF) is an Set theory#Formalized set theory, axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the Type theory, theory of types of ''Principia Mathematica''. Quine first propose ...

or the theory of semisets, 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
In 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 ...

has proper classes which are subclasses of sets.
Notes

References

* * * Raymond M. Smullyan, Melvin Fitting, 2010, ''Set Theory And The Continuum Problem''. Dover Publications . * Monk Donald J., 1969, ''Introduction to Set Theory''. McGraw-Hill Book Co. .External links

* {{Set theory Set theory