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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?).

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I cannot see any difference. Usually call it the underlying set. More often, (wrongly) don't bother to make a notational distinction between a model and its underlying set, letting context determine intended meaning. – André Nicolas May 25 '12 at 3:47
As far as I know, the universe of a model and the domain of a model are exactly the same thing. I double checked in Wilfrid Hodges' 'A Shorter Model Theory' to confirm this. – student555 May 25 '12 at 3:54
@student555: So that the question will not remain unanswered, could you write a brief answer? Clearly it can be very short. – André Nicolas May 25 '12 at 5:05
@AndréNicolas: Is the answer simply, "universe" and "domain" are the same; they both refer to the underlying set of the model. – pichael May 25 '12 at 5:20
@picheal: Yes. By the way, Wikipedia is good for quick overviews, but the words don't always bear close scrutiny, and for detail one should go to less anonymous sources. – André Nicolas May 25 '12 at 5:26
up vote 1 down vote accepted

Universe and domain are the same; they both refer to the underlying set of the model.

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