# Application of model-theoretic forcing

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 $T'$.
• Thank you. Could you elaborate on what you had in mind when you said "first-order [...] techniques on $T^*$"? – Pteromys Feb 4 '18 at 20:46
• I guess I mean something like this. Suppose you have some infinite set of axioms for $T^*$. Let us call it $\Sigma$. Now by compactness if $T^*$ is finitely axiomatisable, then a finite subset of $\Sigma$ would work. In some cases (e.g. fields of characteristic $0$) it is easy to see that this won't work. – Levon Haykazyan Feb 5 '18 at 1:10