Quote by dmuthuk
I haven't studied "classes" yet, but I assume they are pseudosets of a sort that fail to be sets becuase they violate some axiom? For example, is the universal "set" and the "set" of all sets that are not members of themselves classes? And, so the functions and relations that are defined on these classes (which we usually think of as subsets of cartesian products) are now classes as well, I assume.

A class is indeed formally similar to a set. The simplest description is that a class is something defined by a logical formula. For example, the class of singleton sets:
x in SingletonSet iff there exists a y such that x = {y}.
We usually use the setbuilder notation for classes too:
SingletonSet := { x in Set  There exists y such that x = {y} }.
The axiom of replacement works for classes and logical functions, so you can write it in the simpler fashion:
SingletonSet := { {x}  x in Set }.
But note that we cannot carry out Russell's paradox: we can form the class
Russell := { x in Set  x is not in x },
but all we can prove is that Russell is not a set, and thus must be a proper class.