In model theory a structure has an underlying set. In addition to the interpreted relations, are there (implicit) assumptions made about possible operations on this set? For example, is it assumed to have unions, satisfy some of the axioms of ZF, ZFC?
Thanks, John
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As with other branches of mathematics, for basic (naive) model theory, all we need is naive set theory, and the notion of a set is just as naive.
And as with other branches of mathematics, at some point, we do need some formal set theory, so we assume that the sets we are talking about live in the mathematical universe, governed by ZFC. In fact, many model theorists find it convenient to assume, beyond ZFC, that there is a proper class of strongly inaccessible cardinals, though this assumption is usually just a simplification (namely simplifying the notion of a monster model).
The only internal assumption about universes of models in model theory (besides those forced by being a set in the Von Neumann universe) is, as far as I can tell, that they are non-empty. One can do model theory with empty structures, but that forces one to make a number of unpleasant exceptions, while empty structures are clearly beside the point, so it is usually avoided.
No assumptions are made about the elements of the underlying set of a structure in model theory. In fact, the definition of a model doesn't require those elements to be sets (and would work if one's metalanguage has ur-elements or is some kind of type theory where not everything is a set).