Editing Class (set theory) (section)

{{short description|Collection of sets in mathematics that can be defined based on a property of its members}}
In [[set theory]] and its applications throughout [[mathematics]], a '''class''' is a collection of [[Set (mathematics)|sets]] (or sometimes other mathematical objects) that can be unambiguously defined by a [[property_(mathematics)|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 ''{{format link|#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 number]]s, and the class of all sets, are proper classes in many formal systems.

In [[Willard Van Orman Quine|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.<ref>[[Bertrand Russell]] (1903). ''[[The Principles of Mathematics]]'',  [https://archive.org/details/principlesofmath005807mbp/page/n109/mode/2up Chapter VI: Classes], via [[Internet Archive]]</ref> 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|date=November 2024}}