Which classes are sets? - MathOverflow

Which classes are sets?

2010-08-06T15:13:21Z

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.

Answer by Andreas Blass for Which classes are sets?

2010-08-06T15:37:08Z

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).

Answer by Christoph-Simon Senjak for Which classes are sets?

2010-08-07T21:23:34Z

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.