# Class (set theory)

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. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially Russell's paradox (see *§ Paradoxes*). 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 (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 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.^{[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.

## Examples

The collection of all

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 universal class), 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.

## Paradoxes

The

^{[2]}

## Classes in formal set theories

^{[3]}

^{}[4]

^{p. 339}

Semantically, in a

^{[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. However, if an inaccessible cardinal 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. A class function is not a function in the usual sense, since it is not a set; it is rather a formula with the property that for any set there is no more than one set such that the pair satisfies . For example, the class function mapping each set to its powerset may be expressed as the formula . The fact that the ordered pair satisfies may be expressed with the shorthand notation .

Another approach is taken by the

Morse–Kelley set theory 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 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. For example, any set theory with a universal set has proper classes which are subclasses of sets.

## Notes

**^**Bertrand Russell (1903).*The Principles of Mathematics*, Chapter VI: Classes, via Internet Archive**^**Herrlich, Horst; Strecker, George (2007), "Sets, classes, and conglomerates" (PDF),*Category theory*(3rd ed.), Heldermann Verlag, pp. 9–12**^***abeq2 – Metamath Proof Explorer*, us.metamath.org, 1993-08-05, retrieved 2016-03-09**^**J. R. Shoenfield, "Axioms of Set Theory". In*Handbook of Mathematical Logic*, Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)

## References

- ISBN 978-3-540-44085-7
- Springer-Verlag
- Smullyan, Raymond M.; Fitting, Melvin (2010),
*Set Theory And The Continuum Problem*, Dover Publications, - Monk, Donald J. (1969),
*Introduction to Set Theory*, McGraw-Hill Book Co.,