As I understand, Russell's paradox demonstrates that not every class can be regarded as a set. He defines
$$S:=\{x: x \text{ is a set such that }x\notin x\}$$
Assuming that $S$ is a set, this gives a contradiction. However, if in the above definition we replace "set" by "class", we find that $S$ cannot be a class. In other words, the paradox can be used for any structure, not just sets.
My (naïve) understanding is that a set can be identified as a single object, while that's not necessarily true for classes, which can be any collection of objects. If that is true, then in the above definition we could not say things like "$x$ is a class such that…" since it identifies the class as a single object $x$. That would seem to resolve my confusion, but I saw in books sentences like "Let $A$ and $B$ be classes…" which confuse me further, since they again refer to classes (which are not necessarily sets) as individual objects $A$ and $B$.
Surely my reasoning is wrong. What am I missing? What is the difference between a class and a set?
Best Answer
There are lots of ways to paint the fine details, but the broad stroke is:
When you use class-builder notation, such as in $\{ x \mid x \notin x \}$, the notation is only meaningful when $x$ quantifies over sets (or some subclass thereof).
In order to speak of collections of classes, you would need to appeal to some higher object, which we might call a 2-class. And to speak of collections of 2-classes, you'd need a 3-class, and so forth.
For example, one way to make this all precise is by higher order logic. As applied to ZFC, sets would be the elements of the theory, classes would be (first-order) predicates on sets, 2-classes would be second order predicates, and so forth.