# Applications of forcing in model theory

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

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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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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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No one mentioned Shelah's paper "Independence results", in which the notions of proper forcing and oracle are introduced, and the following model theoretic results are proved using them:

(A) There are theories $T⊆T_1$ of cardinalities $ω,ω_1$, respectively, such that $T$ is $ω$-unstable but superstable and $T_1$ has up to isomorphism just one model of cardinality $ω_1$.

(B) An analogous result of $(A)$ for an unsuperstable theory (to prove this, Shelah combines oracles with proper forcing).

(C) Also a model of $ZFC$ is constructed in which $2^ω=ω_2$ and there is a universal linear order of cardinality $ω_1$.

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Fortunately the central notions of first-order model theory are all absolute (among them $\aleph_0$-stability, stability, superstability, simplicity, NIP, deep, dop, otop, $\aleph_1$-categoricity etc). I don't know any MAJOR applications of forcing to first-order model theory. However there are some: Shelah's original characterization of NIP theories used Mitchel's model where Ded$\mu < 2^\mu$ + absoluteness (later a direct ZFC proof was found). Barwise + Kunnen used forcing to establish some results concerning Hanf-Morley numbers. With AECs the situation is different, Shelah established that categoricity in $\aleph_1$ is not absolute, however I am not certain that even this result should be called MAJOR.

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I am not sure if it counts as "set theoretic" forcing, but Makkai's the model existence theorem, which is a uniform treatment of Henkin constructions for $L_{\omega_1, \omega}$ can be viewed as a modification of Robinson's model theoretic forcing. For a good reference see Hodges book "Building Models by Games"

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Here is another (minor) example. Forcing shows up in the area of models of arithmetic, and also of course in the (related) area of models of set theory. The methods of forcing allow one to add a class of a certain kind while ensuring that the resulting model still satisfies a nice theory.

Specifically, I have in mind Simpson's interesting theorem on pointwise definability. A model is pointwise definable if every element is definable without parameters.

Theorem.(Simpson) Every countable model of arithmetic $M=\langle M,+,\cdot,0,1,<\rangle\models\text{PA}$ has an expansion by an inductive predicate $U\subset M$, such that $\langle M,+,\cdot,0,1,<,U\rangle\models\text{PA}^*$ and is pointwise definable.

The predicate $U$ is constructed so as to be generic over $M$, and this is how forcing is involved. See the article, S. Simpson. Forcing and models of arithmetic. Proceeding of the American Mathematical Society, 43:93–194, 1974.

The argument also applies to ZFC. In fact, by coding, one can omit the predicate $U$ by moving to a forcing extension (See Ali Enayat. Models of set theory with definable ordinals. Archive of Mathematical Logic, 44:363–385, 2005). David Linetsky, Jonas Reitz and I extend this in our paper Pointwise definable models of set theory, in which we show that every countable model of ZFC and indeed of GBC has a pointwise definable extension, one in which every set and class is definable without parameters.

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