In set theory and its applications throughout mathematics, 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.
The precise definition of "class" depends on foundational context.
In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.
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.
This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology.
[1] 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.
[non-primary source needed] The collection of all algebraic structures of a given type will usually be a proper class.
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 numbers are a proper class of objects that have the properties of a field.
This method is used, for example, in the proof that there is no free complete lattice on three or more generators.
The paradoxes of naive set theory can be explained in terms of the inconsistent tacit assumption 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 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.
A conglomerate, on the other hand, can have proper classes as members.
is a structure interpreting ZF, then the object language "class-builder expression"
holds; thus, the class can be described as the set of all predicates equivalent to
[citation needed] Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes.
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".
In ZF, the concept of a function can also be generalised to classes.
For example, the class function mapping each set to its powerset may be expressed as the formula
This causes NBG to be a conservative extension of ZFC.
This causes MK to be strictly stronger than both NBG and ZFC.
In other set theories, such as New Foundations 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.