# Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets 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.
A class that is not a set 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 numbers, 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 perhaps rather take place without considering that certain classes can fail to be sets.

## Examples

The collection of all algebraic structures of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects 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.
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 on three or more generators.