One often talks of the cardinality of a model (or more exactly, the cardinality of the universe of the model), such as when one is referring to countable models. Since the universe is, in this model, not a set to which one can assign cardinality, presumably this is referring to the cardinality of this universe when it is embedded in another universe in which the erstwhile proper class becomes a set. So far, so good?
Then one says that a class that has the same cardinality with the universe must be a proper class. For example, the class of ordinals. However, what does one do about the infinite set in a countable model of ZF? The universe and the set are both countable, yet the infinite set is not a proper class.
Sorry for the delay in my response.
From the point of view of another model and another theory, yes. However, if <M,R> |= T, with M being the universe and T being a first-order theory, then the sets designated by T are elements of M, and M itself is not a member of itself. Hence, with respect to T, M is not designated as a set; hence it is a proper class.Usually, the universe for a model IS a set.
I don't know see a rigorous notion at work in "universe M of a model <M R> has property P with respect to theory T".
I'll restrict the context of the following to ordinary set theory (such as Z set theory and its ordinary extensions) and ordinary class theory (such as NBG).
[notation: '0' for the empty set, 'A' for the universal quantifier, 'E' for the existential quantifier, 'e' for 'is a member of']
Start with some definitions:
x is a class <-> (x=0 or Ey yex)
x is a set <-> (x is a class & Ey xey)
x is a proper class <-> (x is a class ~Ey xey)
Now in oridinary set theory we may prove
Ax x is a set
and in ordinary class theory we may prove
Ex x is a proper class.
So, in set theory, when we refer to certain proper classes, that is only informal for schemata in the meta-theory in which we refer to formulas. For example, the locution 'the class of ordinals' refers really to the formula 'x is an ordinal'.
Now, suppose we're using a formal (or even informal) set theory as our theory H in which to do mathematical logic. In H, we define 'model for a language', 'model of a theory', etc. And in so doing, we have no choice but that the universe of the model is a set, because ANYTHING we refer to in that theory is a set.
Now, suppose we're using a formal (or even informal) class theory as our theory J in which to do mathematical logic. Now we may refer to things that are not sets, but still, if I'm not mistaken, the definition of 'model' requires that the universe is a set. Indeed even saying <U R>, where U is the universe, requires that U is a set, since U can't be a proper class while also being the first coordinate of the ordered pair <U R>.
Now, there might be a way to develop models in a class theory so that the universe of a model may be a proper class, but I don't happen to know how it would be done. Also, I don't know how one would express and then prove things (such as the completeness theorem) using models with universes that may be a proper class.
What I've said also does not contradict the informal notion of proper classes as universes for proving such things as relative consistency theorems. For example, using L (the constructible universe) and things like that. Again, in such cases, this resolves to mention of formulas (such as relativization of formulas to a "proper class" as the "proper class" is not itself an object but rather a locution for the relativizing formula "x is constructible").
Finally, as to your notion of a universe being a proper class with regard to a theory T, recall that a theory is syntactical and a consistent theory admits of lots of DIFFERENT models. So while the universe U of a given model of T is not a member of itself (given the axiom of regularity), it is not precluded that that U may be a member of a different universe for a different model of T. And, back to the original point, the cardinality of U is not affected by whether it is or is not a universe of any given theory T.
In sum, where we use either set theory or class theory as our theory in which to do mathematical logic (including the subject of models), a proper class does not have a cardinality, while every set (with the axiom of choice) does have a cardinality equinumerous with that set. For example, suppose w (the set of natural numbers) is the universe of some given model. The fact that w is not in w does not in the least detract from the theorem that w has a cardinality, and irrespective of any theory of which w may be the universe of a model of that theory.