Collections, classes, sets

Question

( 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.

Questions:

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?

Quite right; in axiomatic set theory, a class is a collection that is not a set (e.g. "too large", etc.) — Mauro ALLEGRANZA, Feb 06 '18 at 14:44
Related post: Morse-Kelley class comprehension and Russell's paradox and Notation for the set of all sets as well as Representation of classes by sets in Bernays's set theory. — Mauro ALLEGRANZA, Feb 06 '18 at 14:45
The set of all sets leads to paradoxes. A class is like a set, but with slightly different axioms so that having things like a class of all sets doesn't cause paradoxes. — Jonathan, Feb 06 '18 at 14:49
Also, set and class have widely accepted definitions in mathematics, but collection does not. So collection usually has the same vague meaning it does in non-mathematical English. — Jonathan, Feb 06 '18 at 14:56
Kelley doesn't talk about ZF, so if you try to infer ZF from Kelley, you won't get to the right place. There are significant differences. For example, in ZF, every class is a set, but in MK, there are "proper" classes that are not sets. One example is the universal class, which simply doesn't exist in ZF. — MJD, Feb 06 '18 at 15:05
To your edit, https://math.stackexchange.com/questions/1452425/what-is-the-definition-of-a-set/ — Asaf Karagila, Feb 06 '18 at 18:31
It seems to me, also, that you are trying to ask "what is a boson? What is baryonic matter? What is dark matter?" without understanding any physics. (This is an analogy!) The truth is that what are sets and classes depends on your context, and it will change from one work to another as the context changes. Much like the term "number" will change between contexts. There is no "concise answer" which is "completely correct" and at the same time "to the point". — Asaf Karagila, Feb 06 '18 at 18:35

score 2 · Accepted Answer · answered Feb 06 '18 at 15:01

Collection is not being used as a mathematical term at all; it is an informal, English word that doesn't really have any mathematical connotations, so it often gets used in English descriptions as a means of avoiding potential confusion.

The actual mathematical terms here are set and class.

It is possible to talk about "collections of classes"; if you identify classes with the sorts of things you can describe using first-order logic, then collections of them could be expressed using second order logic.

More commonly, however, when one wants to speak of such things one leverages set theory; e.g. with a large cardinal axiom you can have a universe of small sets (which, nonetheless, satisfy the axioms of ZFC; small is only being used as a relative term), and then identify proper classes with certain kinds of large sets. Then, you can talk about sets of classes with ordinary set theory.

Hmm So what is the undefined thing here? I mean, we can't define everything, we have to stop somewhere: I thought that that was the role of "collection" - the same role that I think that set has in naive set theory... — soap, Feb 06 '18 at 15:07
@Soap: "Set" is the undefined term in ZFC. (and if we're doing ZFC in first-order logic, "Class" isn't a part of the language: it's referring to logic) — , Feb 06 '18 at 15:13

score 2 · Answer 2 · answered Feb 06 '18 at 15:11

One comment. It really is a bit unhappy to say "A Set is a collection respecting the axioms of ZFC set theory."

There are theories of sets other than ZFC! There are, for instance, set theories like NFU which do allow a universal set (a set of all sets). And there are more conventional but still distinct set theories like Scott-Potter which aim to capture the picture of cumulative hierarchy of sets better than traditional ZFC.

Don't be too swayed by ZF-iste imperialist ambitions! (For some of the options see very briefly §7.3 of this Study Guide, and for more see Randall Holmes on Alternative Axiomatic Set Theories.)

I agree! would you then define set as a collection that satisfies one of those sets of axioms? — soap, Feb 06 '18 at 18:21

Collections, classes, sets

2 Answers2