I have a universal first order universal theory T such that:
-T is universal over a finite language L
-T has a model completion T ⊆ T*
-T* is countable, has quantifier elimination (QE) and is complete.
Now considering a theory T ⊆ T' over L, suppose T' has QE. Can we say that T' is (at most) countable? If so, are there any assumptions I can drop and T' will still be countable? Alternatively, if not, are there any assumptions I could add to guarantee T' will be countable?
Thanks.