# What is the significance of “classes”?

In the introduction of Hungerford's Algebra (p. 2), he gives a rather trivial example of a class that is not a set, but what is the purpose of even having this term defined? Is it useful, other than to give a name to collections of objects that are not sets?

Also, how is this term related to equivalence and congruence classes? More specifically, are there equivalence or congruence classes that are not sets?

EDIT: It turns out the "rather trivial example" is Russell's paradox and wasn't so trivial at the time of it's discovery.

• This is more an informal remark than an answer: for many if not most professional users of mathematics, the term "class" is indeed only used as a way to speak of collections of objects that are too big to be sets (if you like, just a near-synonym for the informal term "collection" that has the value of signaling that the collection referred to might not be a set). It does have a precise role in specific formalizations of mathematics (see JBeardz' answer), but many people using the term do not have the details of this or any other formalization in mind when they use it. – anon Mar 6 '11 at 5:13
• This Discussion actually answers my question perfectly! – dandiellie Mar 7 '11 at 18:50

## 2 Answers

There is no connection to equivalence/congruence classes - that's just a coincidence.

Classes are useful since sometimes we want to talk about them as sets. For example, instead of saying "for every ordinal $\alpha$", you can say "$\forall \alpha \in \mathrm{On}$". In other words, it's just for convenience.

• There is almost an overlap: in many categories, the notion of isomorphism would be an equivalence relation, but for the fact that it doesn't fit in a set. The informal mathematical practice of ignoring this, e.g. talking about isomorphism as an equivalence relation and referring to the "isomorphism class" of a group or "homeomorphism class" of a topological space without specifying a reference set on which the isomorphism is a set-theoretic equivalence relation, just (coincidentally) happens to be completely in line with formal use. (These things are proper classes). – anon Mar 6 '11 at 5:25

Classes were introduced so that we could have a "collection" of things which is not a set, for instance the set of all sets (does it contain itself...?). A class that is not a set is called a proper class, e.g the "class" of all sets. It's just sort of a sneaky way of avoiding paradoxes. See http://en.wikipedia.org/wiki/Proper_class for more info.