# Applications of forcing in model theory

What are the major applications of (set theoretic) forcing in model theory?

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Rather than addressing the issue of whether an application is major or not, let me point out a recent one: See

John Baldwin, and Paul B. Larson. Iterated elementary embeddings and the model theory of infinitary logic, preprint.

(The paper is downloadable from either of their webpages). What they do is work in the context of Abstract Elementary Classes, and provide a unified framework for several results on the number of models of size $\aleph_1$ in various infinitary logics. Their approach uses "$\mathbb P_{\mathrm{max}}$-techniques" to obtain the models through iterations of countable transitive models of enough set theory via embeddings given by the non-stationary ideal on $\omega_1$. The models so produced are typically ill-founded, as the technique is applied without the assumption of large cardinals.

Similar constructions appear in the follow-up paper

John Baldwin, Paul B. Larson, and Saharon Shelah. Almost Galois omega-stability, preprint.

(Also available at their websites.)

Actually, recent work on Abstract Elementary Classes provides a nice environment for non-trivial interactions between model theory and set theory, not just in the form of forcing techniques. See for example this post at the Forking, Forcing and back&Forthing blog.

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Applications of forcing in model theory, following directly from Robinson's 1970 work, are reviewed by T. Zhang, Model-theoretic Forcing and Its Applications.

For a more recent application, the construction of finitely generated models, see A.O. Houcine, Forcing in Model Theory revisited.

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