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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 Discussion](http://math.stac$k$exchange.com/questions/24507/why-did-mathematicians-take-russells-paradox-seriously) actually answers my question perfectly!2011-03-07

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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.

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    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).2011-03-06
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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.