Class

A class is a collection of different things with Propositional equality which may not be a set, though the exact meaning depends on the foundation being used. Thus it is a generalization of a small collection. A proper class is a class which is not a set. A subclass generalizes a subset. A mapping between classes is a class function.

Foundation-agnostic usage

We often use the word class in foundation-agnostic contexts. The appropriate interpretations are then:

In a theory like $Z F$ one may treat a class indirectly as a predicate $Φ$ ranging over sets and possibly urelements and we say $𝑥 \in Φ$ iff $Φ (𝑥)$ .
In an extension such as $N B G$ , 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 $ℭ 𝔩 𝔰 (𝑥)$ . Again, classes are taken to satisfy the Axiom of Extensionality
In $T G$ a $U$ -class is a subset of $U$ .
Similarly, in a Type theory with universes a class is the same as an h-set.

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