Class (set theory)

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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.

The precise definition of "class" depends on foundational context.
In work on Zermeloโโ‚ฌโ€œFraenkel set theory, the notion of class is informal,
whereas other set theories, such as von Neumannโโ‚ฌโ€œBernaysโโ‚ฌโ€œGรถdel set theory,
axiomatize the notion of "proper class",
e.g., as entities that are not members of another entity.


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.

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A class that is not a set (informally in Zermeloโ€“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.

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Another approach is taken by the von Neumannโ€“Bernaysโ€“Gรถdel axioms (NBG);
classes are the basic objects in this theory, and a set is then defined to be a class
that is an element of some other class.

However, the class existence axioms of NBG are restricted so that
they only quantify over sets, rather than over all classes.
This causes NBG to be a conservative extension of ZF.