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I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem to construct an analogue of set-theoretic forcing in Model Theory. This was very interesting to me as it was the first time I had heard this.

I am extremely interested in the $\forall_2$ theory $Art_{k}^{l}$ of local artinian rings of length at most $l$ containing the field $k$ in the language of rings. To get an application, we must insist on the language being countable, which will amount to requiring that the field $k$ is countable.

My question is the following: what are the enforceable models the theory $Art_{k}^{l}$. This means what models $M$ admit the enforceable property of "$M$ is isomorphic to the compiled structure" ?

In particular, can we use this to say something about $Art_{\mathbb{Q}}^{l}$ and/or $Art_{\mathbb{F}_q}^{l}$?

Forgive me, if the terminology is a bit wonky. I am asking the question because I am not familiar with forcing in model theory and much less forcing in set theory. Hodges' "Model Theory" is the only reference I have; however, I know that the omitting types theorem is a fundamental result in model theory. I hope someone is familiar with this construction.

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ah, I should mention the following fact: enforceable models are existentially closed. The existentially closed models of local aritinian rings of length at most $l$ containing the field $k$ are Gorenstein aritinian rings of length $l$ containing the algebraic closure of $k$. This is per H. Schoutens' work on existentially closed models of local aritinian rings. Does requiring the model to be enforceable and not just existential closed, give us even more properties? –  Andrew Stout Mar 10 '13 at 13:33
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