Distinction between the universe of a model and the domain of a model?
I'm pretty sure I'm wrong about this. But even reading Wiki, I'm still not clear.
I'll use an example to illustrate what I take to be the distinction:
It seems to me a model can have a finite domain but an uncountable universe: {R}. The domain has 1 element, while the universe has uncountably many elements.
EDIT: Isn't there another difference? For example, we might say that in order for M to be a countable model of ZFC there must be universe, V (otherwise from what perspective could we say that M was countable?). Here, domain and universe are different. Furthermore, is V a proper class (the class of all sets?).