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From Smullyan and Fitting's Set Theory and the Continuum Problem:

Which classes are sets? Rather than attempt an absolute answer to this (which some authors have done with dubious success), we regard it as philosophically more honest to take these notions as only relative to any given model of the axioms of class-set theory.

This is fine for the text, but it makes me curious what attempts have been made at absolute answers, dubious or not.

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The attempts to give absolute answers tend to get formalized into theories, and from then on they can be regarded as being relative to the axioms of that theory.

For example, Zermelo-Fraenkel set theory can be based on the "absolute answer" that a class is a set if and only if the ranks (in the cumulative hierarchy) of its members are bounded above, so that there is a level of the cumulative hierarchy at which all those members can be collected into a set.

Similarly, Quine's "New Foundations" uses the idea that a class is a set if (and only if? --- I'm not so sure about this converse) membership in the class can be specified by a stratified formula.

There is a less known but very interesting set theory due to Wilhelm Ackermann (Zur Axiomatik der Mengenlehre. Math. Ann. 131 (1956), 336--345) in which a class is a set if membership in it can be specified by a condition that does not mention the general concept of set (and that uses only sets as parameters).

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How does Bourbaki's definition of a set using coll_x and the tau operator compare? –  Harry Gindi Aug 6 '10 at 15:47
While I appreciate this answer (incidentally, Smullyan and Fitting's book uses NBG) the above quote suggests some think there is an overarching principle that connects all possible definitions. Even if it's wrong as posited, I'm curious what that principle might be. –  Jason Dyer Aug 6 '10 at 16:10
I wouldn't say that Quine's approach in NF is "similar", though there's ambiguity in that term. Using a direct restriction on the comprehension principle to dodge the paradox is a fundamentally different approach from the limitation of size approach. Also, I hadn't thought that NF even relies on a set/class distinction in the first place. Does it? –  Jeremy Shipley Aug 6 '10 at 16:23
It's true that neither NF nor ZF involves classes, but the original question presupposed them. In both cases, to make sense of the question, one should interpret classes as "virtual" in Quine's sense, i.e., as informal collections $\{x:\phi(x)\}$ of sets satisfying some property $\phi$. And in both cases, the axioms are usually formulated as saying that certain sets exist rather than that certain classes are sets. In both cases, there are extensions incorporating classes explicitly into the theory. Ackermann's theory, in contrast, uses classes from the start. –  Andreas Blass Aug 6 '10 at 18:59
@Harry: I won't try to explain Bourbaki's notion of set, partly because I don't remember those axioms precisely enough, and partly because Bourbaki's attitude seems to be far more formalistic than what is presupposed in the question. I believe, though, that Bourbaki's set theory can be viewed as describing a notion of set similar to that of ZF but with (here's the part I don't remember well enough) some restriction on the height of the cumulative hierarchy (perhaps just height $\omega+\omega$). –  Andreas Blass Aug 6 '10 at 19:02
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It may not completely fit, but - just to have said that - in Zermelo-Fraenkel Set Theory, there are no classes. Classes are a concept of the meta-theory - a "class" is no object inside any model of ZF, it is a collection of sets that can be described somehow, mostly by a proposition with one free variable $A(x)$.

Its a convenient way to write down even the axioms of ZF, but you cannot quantify over them, and every proof of the form "for all classes ..." I know is just by structural induction on formulas.

ZF itself doesnt say anything about classes directly (except maybe that some are not empty). It specifies a few simple given objects and operations you can do on them to create more complicated objects, and these objects are called sets.

Actually, I dont see the point in such discussions. I mean, the reason for not having general comprehension $\{x|A(x)\}$ is that you run into the well-known contradictions (like $\{y|y\not\in y\}\in\{y|y\not\in y\}\rightarrow\{y|y\not\in y\}\not\in\{y|y\not\in y\}$), so one has to restrict the notion of a "set" so far that he still can do anything he needs, but doesnt produce contradictions. And then classes are kept as a concept for collections of sets that cannot be expressed in this notion, when you need them. I dont see the "philosophical" consequences of this.

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