For n an integer greater than 2, Can one always get a complete theory over a finite language with exactly n models (up to isomorphism)?
There's a theorem that says that 2 is impossible.
My understanding is this should be doable in a finite language, but I don't know how.
If you switch to a countable language, then you can do it as follows. To get 3 models, take the theory of unbounded dense linear orderings together with a sequence of increase constants < ci: i < ω >. Then the ci's can either have no upper bound, an upper bound but no sup, or have a sup. This gives exactly 3 models. To get a number bigger than 3, we include a way to color all elements, and require that each color is unbounded and dense. (The ci's can be whatever color you like.) Then, we get one model for each color of the sup plus the two sup-less models.