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 $T'$.

  • $\begingroup$ Thank you. Could you elaborate on what you had in mind when you said "first-order [...] techniques on $T^*$"? $\endgroup$ – Pteromys Feb 4 '18 at 20:46
  • 1
    $\begingroup$ 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. $\endgroup$ – Levon Haykazyan Feb 5 '18 at 1:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.