( Edit: I understand that a very similar question has been asked before, but in this case I was hoping for: confirming that I was thinking correctly; and get a more clear/concise answer if possible by doing so, since the answers to the other questions about this, although very interesting, seemed to not clearly state: the definition of set is this, and of class is this, and of collection is this)
I am trying to understand the distinction between set, class and collection. I don't know that much about logic, although I am familiar with concepts like first order languages and universes, but not much more than that. By reading other posts here and the appendix of Kelley's General Topology, this is the idea that I got:
- Collection is left undefined - it is assumed that we know what it means: a "bunch" of "things". It is the most general concept of these (and maybe of mathematics??)
- A Set is a collection respecting the axioms of ZFC set theory.
- A Class is a collection of sets
So in particular a set is a class which is an element of another class, and a class that is not a set is called a proper class. A class that is a set is called a small class. We can now for example talk about the class of all sets, although it must be a proper class, since otherwise we get Russel's paradox.
- How much of this is incorrect?
- Can we talk about the "collection of all classes", which will contain in particular the class of all sets? Do some paradoxes arise?
- Any other things to add to this list or maybe some enlightening comments?