# 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.

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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

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.