Russell's paradox and ZFC, "set" vs. "class" I am reading the chapter on "Set Theory" in "Philosophy of Mathematics" by George and Velleman.
I suppose questions similar to this have been asked many times, but I couldn't find anything satisfying, so I'm asking anyways.
As is well known, "the set of all sets that don't contain themselves" does not exist, and therefore not every concept has an extension as Frege envisioned. Those sets which don't contain themselves are then often called "collection" or "class".

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*How does calling them a different name improve anything? I assume that those words are just euphemisms for saying "doesn't exist." I tend to think of a class as a 'method to collect objects' (or just sets, if thinking in ZFC terms), which may be thought of as progressing through the stages of set-construction (as in ZFC), but never 'converging' to a whole. This being opposed to a set which is assumed to be a whole, existing prior to construction. ("Construction" seems the wrong word -- "description" may be more appropriate).

*Doesn't the fact that Russell's predicate leads to a contradiction imply that the objects satisfying the predicate cannot be 'collected' into a whole, i.e. that there is something wrong with the predicate (being paradoxical), as opposed to the entire theory? (Maybe "cannot have been collected" is a better way to put it.)

*Would it not be simpler to reject self-contradictory predicates / concepts not allowing extension as ill-formed and move on with Frege's theory, instead of trashing it all together? Indeed, in light of 1., a definition of a set as "gathering into a whole of definite, distinct objects..." a la Cantor seems to exclude proper class concepts such as Russell's, since they're merely methods of collection, not existing, definite objects.

What am I missing? Maybe the existence of extensions for all concepts is truly required for Frege's theory?
I assume that this is a common objection every student raises, so if there is a discussion somewhere which addresses this, please do link it, since I was not able to find it.
Thank you!
 A: The metalanguage of ZFC allows arbitrary collections of sets (one per unary predicate), which in general we call classes. "The class of all sets with a given property" may or may not be a set, but if its being a set would imply a contradiction, that proves it's a class that isn't a set, aka a proper class. It wouldn't be "simpler" to reject the metalanguage's contents.
We say a class "is" a set when some set in ZFC has precisely the same elements as that class; in that case, we identify that class with said set. Both of these linguistic conventions are strictly speaking abuses of notation, but that's OK; mathematics is full of such abuses, otherwise we wouldn't have a technical term for them.
One consequence is that a class is a set iff it's an element of a class. So while set theory was originally invented to formalise a notion of collections, ironically sets are defined by being elements rather than things that having them. (ZFC lacks urelements; if it had some, this issue would be more complicated.) In particular, the Russell class exists, is a class of sets (in fact, in ZFC it's the class of all sets, aka the universe), but isn't a set so isn't an element of itself. This isn't paradoxical, because an arbitrary class is only defined by the condition for a set to be an element.
In some set theories that might be better called class theories, the class is fundamental and in the object language, and a set is defined as a class that is an element of some class. Axioms therein often make explicit a focus on sets.
A: Let me add one remark concerning your third point.

Would it not be simpler to reject self-contradictory predicates / concepts not allowing extension as ill-formed and move on with Frege's theory, instead of trashing it all together?

The problem is that we do not know how to detect self-contradictory predicates in general. It would be very nice to have a philosophically meaningful criterion telling us which predicates define sets (i.e., which predicates are consistent) and which don't.
Quine's $NF$ tries to provide such a criterion by the notion of stratification; decide for yourself how convincing it is.
$ZFC$ on the other hand evades the definition of such a criterion. It rather states a lower approximation to the concept of a consistent predicate: Certain small collections are declared sets, and then by applying set-theoretic operations (union, powerset etc)
more and more collections are declared sets. At a certain point, enough sets have been collected to formalize mathematics. And on the other hand, not too many collections have been made into sets: It seems like we cannot derive contradictions such as Russell's in $ZFC$ (note however that we cannot be sure about the latter point, because we do not know if $ZFC$ is consistent).
In light of the technical success of $ZFC$ - it was good enough to provide mathematicians with all the sets they needed while appearently avoiding paradoxes - the task of finding a general answer to the question which classes are sets lost its urgency, at least for the "working mathematician".
