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, i ...

and its applications throughout mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property 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. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...

(See #Paradoxes). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory
In set theory
illustrating the intersection of two sets
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 ...

, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory
In the foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theorie ...

, 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
Set theory is the branch of that studies , 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 , is mostly concerned with those that ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

s, the class of all vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, and many others. In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

, a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

whose collection of objects
Object may refer to:
General meanings
* Object (philosophy)
An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...

forms a proper class (or whose collection of morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s forms a proper class) is called a large category
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

.
The surreal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s are a proper class of objects that have the properties of a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

.
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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 act or change without constraint
* Emancipate, to procure political rights, as for a disenfranchised group
* Free will, control exercised by rational agents over their actions and decis ...

complete lattice
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

on three or more generators.
Paradoxes

Theparadoxes of naive set theory
A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-con ...

can be explained in terms of the inconsistent tacit assumption
A tacit assumption or implicit assumption is an assumption that underlies a logical argument
In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable ...

that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proof
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

s 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. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...

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 of ...

suggests that the class of all ordinal numbers
In set theory
Set theory is the branch of that studies , 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 , is mostly concerned with those that ...

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
Conglomerate or conglomeration may refer to:
* Conglomerate (company)
* Conglomerate (geology)
* Conglomerate (mathematics)
In popular culture:
* The Conglomerate (American group), a production crew and musical group founded by Busta Rhymes
** Con ...

, 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 theoryZF, 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 Factory" ...

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
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents sta ...

, the classes can be described as equivalence classes
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

), and its subsets can be thought of as "classes".
In ZF, the concept of a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

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 In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical pr ...

of ZF.
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 logic, first-order axiomatic set theor ...

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 axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the Type theory, theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New ...

or the theory of semiset{{distinguish, Semialgebraic set
In set theory, a semiset is a class (set theory), proper class that is contained in a Set (mathematics), set.
The theory of semisets was proposed and developed by Czech Republic, Czech mathematicians Petr Vopěnka an ...

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
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...

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