Set, class, family?

Set, family, and class; what are they exactly?

A set is a well-defined collection of distinct objects. But not every collection is a set; we will demonstrate an example. In set theory, the comprehension principle states that for a predicate P(x) formed by first-order logic, there exists a set including all objects that satisfy P(x). But an unrestricted form of this principle leads to what’s called Russell’s paradox. We see that letting P(x): x\notin x, where x is a set, and let R=\{x:P(x)\}, the principle asserts that R exists. It can’t be that R\in R, so R\notin R, which implies that P(R) is true and R\in R. But this leads to a contradiction. But in ZFC, which is the canonical axiom system, an Axiom of Separation was imposed to avoid Russell’s paradox. This axiom essentially restricts the comprehension principle, by that using the set construction method, one can not build a set bigger than the previously constructed sets.

Practically, a collection such as R above, which is not a set, is an example of a proper class, while sets are exactly objects that are known as small classes. In category theory, a category whose collection of objects forms a proper class is called a large category. Lastly, what is a family?

A family of sets is a collection of sets (with no restrictions). That is, a collection with repeated sets or a proper class of sets pass as a family of sets.