Class (set theory)

Collection of sets in mathematics that can be defined based on a property of its members

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.

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.

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 ZF.