I’ve recently run into two arguments that there is no single intuition behind the notion of set. One was a talk by Tatiana Arrigoni in Berlin that I mentioned earlier, arguing that there may be multiple intuitions underlying set theory. Another was a talk by Jose Ferreiros at the Berkeley Logic Colloquium last week, arguing that there isn’t even one intuition underlying the notion of set.
Ferreiros started by pointing out that the 19th century founders of modern set theory (Cantor, Frege, and Dedekind) all seemed to be working with different notions. In particular, Cantor seemed to never consider a set as an element of a further set (his famous theorem was apparently proved about the set of functions on a given set, rather than using the modern notion of a power set). In addition, recognizing some paradoxes that would result from this theorem, he rejected the notion of a set of everything. His notion was fairly close to the intuitive notion of “collection” that we often introduce students to set theory with.
Frege (and Russell, following him) thought of sets as extensions of concepts, or as some way of dividing the universe into two parts. It’s a more “top-down” notion than the others, because of the sort of impredicativity involved in dividing a totality (including the set to be defined) into two parts. This notion is probably closest to our intuitive notion of “property” that the set concept seems to inherit some of from the separation axiom.
Dedekind had a slightly different conception from either. I didn’t quite get the details, but it sounds like it was the only notion that was relatively close to the modern “iterative conception” championed later by Gödel (among others).
The fact that our modern notion of set has inherited properties from each of these suggests that it is not a rigorization of any of these intuitive concepts. He also gave some other arguments, based on the fact that we have intuitions about the existence of “absolutely arbitrary” subsets of any given set, which aren’t successfully cashed out in any of our axioms, but are approximated by the axioms of separation and choice.