[[Set theory MOC]]
# Class

A **class** is a [[collection]] of different things which may not be a [[set]],
though the exact meaning depends on the [[axiomatic set theory]] being used.
Thus it is a generalization of a [[small set]].
A [[Proper class]] is a class which is not a set.
A [[Subclass]] generalizes a subset.
A mapping between classes is a [[Class function]].

In a theory like [[ZF]] one may treat a class indirectly as a [[predicate]] $\Phi$ ranging over sets and possibly [[urelement|urelements]]
and we say $x \in \Phi$ iff $\Phi(x)$.
In an extension such as [[NBG]], a class becomes an object in its own right which largely supersedes the notion of a set,
with a set becoming a class which is contained in some class.
Classhood of $x$ is denoted by $\chood(x)$.
Again, classes are taken to satisfy the [[Axiom of Extensionality]]

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