My question is regarding model-theoretic forcing developed by Robinson.
I understand that model-theoretic forcing is useful in the study of existentially closed models, such as constructing e.c. models and calculating resultants of quantifier-free formulas in groups.
Can it be used to study other aspects of e.c. models? In particular, can it be used to show that a finite subset $T'$ of $T^*$, the model-completion of a universal theory $T$, does not axiomatize $T^*$? Here, I would imagine that one constructs a non-e. c. model satisfying $T'$.
I'm trying to read Hodges's Building Models by Games, but I thought that I should learn if the subject of the book is relevant to the said question that I had in mind.
In Hodges' presentation of forcing two players play a game the end result of which is a structure. At least in the finite forcing, the property of being existentially closed model of $T'$ is enforceable. This means that both players can ensure that the resulting structure is an existentially closed model of $T'$. So that kind of forcing cannot be used to construct a non-e.c. model of $T'$.
My first instinct is that techniques of first-order model are more powerful. So if you have a model completion $T^*$, it is best to use these techniques on $T^*$, rather than try to construct a non-e.c. model of $T'$.