Editing Class (set theory) (section)

== Examples ==
The collection of all [[algebraic structure]]s of a given type will usually be a proper class. Examples include the class of all [[group (mathematics)|group]]s, the class of all [[vector space]]s, and many others. In [[category theory]], a [[Category (mathematics)|category]] whose collection of [[Object (category theory)|objects]] forms a proper class (or whose collection of [[morphism]]s forms a proper class) is called a [[large category]].

The [[surreal number]]s are a proper class of objects that have the properties of a [[field (mathematics)|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 number]]s.

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 lattice#The complete free lattice|free]] [[complete lattice#Free complete lattices|complete lattice]] on three or more [[Generator (mathematics)|generators]].