TITLE: Is the "domain of discourse" in axiomatic set theory also a "set"?
QUESTION [10 upvotes]: The domain of discourse is defined by Wikipedia as the "set of entities over which certain variables of interest in some formal treatment may range." However, I believe we could not call the domain of discourse in axiomatic set theory a "set," since defining the "set of all sets" leads to Russell's paradox. Is it better to consider the domain of discourse to be an intuitive "collection?" Or a formal "class?"

REPLY [2 votes]: 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)