Is the "domain of discourse" in axiomatic set theory also a "set"?

Solution 1:

It is even better not to think of 'the domain' as a single object at all (whether 'set', 'class' or 'collection'). Instead of talking of a domain (singular) of individuals, just talk directly of the relevant individuals (plural). As logicians very often do when they forget about their official story! Thus even the great Alonzo Church talks of the range of the individual variable as "the individuals" [nb the plural!]: when interpreting first-order arithmetic he says "the individuals [plural] shall be the natural numbers ..." and so on (Intro. to Math. Logic, p. 174).

So: use plural talk rather than set talk in your informal metatheory. What are the variables of first-order set theory supposed to run over? The sets [plural] of course; what else?! In this case, the individuals the quantifiers run over can't form a set. But that's ok: there's no need to think of the domain of discourse as a set [singular] rather than as many objects, and indeed the example of set theory shows that we can't always think of a domain of discourse as a set.

If you want to formalize the metatheory in plural style, use a formal plural logic. For one modern account of how to do this, see Alex Oliver and Timothy Smiley Plural Logic (OUP). See also http://plato.stanford.edu/entries/plural-quant/#SetThe

Solution 2:

One seeming-paradox is that any consistent first-order theory has a denumerable model. Therefore, the "sets" in at least some models of Zermelo–Fraenkel set theory can indeed be placed into a set. It's just that the set containing the "sets" in the model cannot be described in the formalization of the theory. For details, see the link.

In other words, we have two concepts of "sets": those that can be described in the formal theory, and those that cannot but can be described in some "larger" theory. Yes, this is weird, but so is formal set theory.

Solution 3:

Well, there is more than one axiomatic set theory.
The domain of discourse of an axiomatic set theory is a collection of objects that may or may be not a set of the axiomatic set theory in question;
In the case of ZFC, the collection of all its sets isn't one of the sets of ZFC, but it could be a set according to some other theory.
To clarify;
Let's prefix the name of the theory to the word set, depending on what we're talking about.
A ZFC-set of all ZFC-sets doesn't exist.
But there could be a theory X so that the collection of all ZFC-sets is an X-set.
Also, there could be a theory Y such as the collection of all Y-sets is a Y-set! I mean, there are such theories (http://en.wikipedia.org/wiki/Universal_set#Theories_of_universality)

So in short, the problem is that a collection of objects is not instantly "a set", because "set" alone isn't enough to specify what you are talking about, unless you somehow create a theory Z that considers Z-sets anything that could be concieved as a collection of objects and decide to call "set" every Z-set.
ZFC is not such a theory, since the collection of all ZFC-sets is not a ZFC-set.
But all ZFC-sets form a collection of objects, just not a mathematical one until you find a theory capable of calling it set according to its terms.
(EDIT: the last statment is wrong, since "class" is a mathematical term too, and that too can describe a collection of objects "mathematically". For instance, in NBG, there is an NBG-class of all NBG-sets. so in NBG, the domain of discouse happens to be an NBG-class. Also note that i don't think X-set is common notation (i could be wrong), it just seemed useful to illustrate the point)