Currently, I am reading Algebra: Chapter 0 by Paolo Aluffi with the initial purpose of learning algebra rather than set and class theory. I can work quite good with naive set theory; but I also understand the Russel paradox, which shows that there cannot be a set of all sets that don't contain themselves as elements (and thus that there cannot be sets that contain all sets, or all vector spaces, or all groups, and so on). Concerning this issue, the author writes:
The reader will note that I refrained from writing a set of objects, opting for the more generic 'collection'. This is an annoying, but unavoidable, difficulty: for example, we want to have a `category of sets', in which the 'objects' are sets and the 'morphisms' are functions between sets, and the problem is that there simply is not a set of all sets15. In a sense, the collection of all sets is 'too big' to be a set. There are however ways to deal with such 'collections', and the technical name for them is class. There is a 'class' of all sets (and there will be classes taking care of groups, rings, etc.).
Thus: Collections of objects of our set theoretic universe of which we know that they aren't sets, we just call classes. I can work quite well with this approach (at least I can do all the exercises)––but I have the feeling that I don't really understand the "set–class issue"!
Let me explain why: Naive set theory gives me an intuitive model of our universe of sets: a set is any collection of objects, where "object" can me urelement (numbers, pairs, functions, everything that isn't a set) or set (i.e., sets are also considered to be a type of object). Then I learnt that this intuitive model of set theory is contradictory, in particular, the idea of the set of "all sets" (or, more generally: the set of "all [insert large collection of strucutres here]") is contradictory. So we know that our original conception of what a set is is contradictory. But what is our new conception of what a set is? Also, how can we be sure that if we have a set X of which we are quite sure that it isn't contradictory, we can replace each element of X by any non-contradictory set and the union of these sets is also non-contradictory?
My problem is that although I can intuitively tell you whether a given collection is a set or "too big to be a set", I have the feeling that I don't understand the full story. I have the feeling that one doesn't need to know much about "set–class issue" in order to do algebra. But isn't this a missed opportunity? I want to understand the full story!