# What are some applications of model theory?

In an attempt to "broaden my horizons", I am taking a class on model theory, which follows this book:

Skimming through the chapters and reading wikipedia, I now have some slight idea of what it is about. However, I am still not clear on its applications (though application is not the primary reason I am taking this class).

Could someone enlighten me on why this field is needed, and as a matter of opinion, what are some of its potential application going forward?

• Model theory is useful enough in linguistics that you can find it discussed in textbooks, like Partee, Meulen, and Wall's "Mathematical Methods in Linguistics": books.google.com/…. I would love it if someone better-versed in linguistics could post an answer with more details. – Vectornaut Jan 18 '15 at 4:50
• mathoverflow.net/questions/7018/… – Martín-Blas Pérez Pinilla Jan 18 '15 at 18:30
• – tomasz Jan 20 '15 at 0:03

If by applications you allow applications to other mathematical field, then here two examples.

The first one is Ax's theorem. It states that a polynomial function $\mathbb C^n \to \mathbb C^n$ is injective if and only if it is bijective. Basically, the proof is as follow : it is clearly true for (polynomial) functions $\mathbb F^n \to \mathbb F^n$ for any finite field $\mathbb F$ ; the Nullstellenstatz then implies that is also true for polynomial functions $\overline{\mathbb F}^n \to \overline{\mathbb F}^n$ with $\overline{\mathbb F}$ the algebraic closure of a finite field ; then comes the model-theoretic argument with Los's theorem which shows that it is again true in the ultraproduct of all algebraic closure of finite field. Then it can be seen that this ultraproduct is an algebraically closed field with characteristic $0$ and infinite transcendence degree over $\mathbb Q$, that is $\mathbb C$.

The second example is the Nullstellensatz itself. It is an immediat corollary of the model-completeness of the theory of algebraically closed fields.

One of the main areas of applied model theory is its application to Algebraic Geometry. Hrushovski proved the geometric Mordell–Lang conjecture for all characteristics by way of model theory.

Another example is Zilber's work on Schanuel's conjecture. I also know that van den Dries has proved theorems about real analysis using O-minimality. If I can find a good example, I will update this later.

(Also, I would like to remark that there are two competing sologans for the field of model theory. On one end, we have classification theory and a slogan witch reads "Model Theory = logic + universal algebra" (This is attributed to Chang and Keisler). More recently, and on the other end of the field, we have another slogan, "Algebraic Geometry - Fields = Model Theory" (This is attributed to Hodges).)

• Congrats about Notre Dame. – Asaf Karagila Feb 2 '15 at 13:59
• @AsafKaragila: Thanks so much! How did you know? – Kyle Feb 2 '15 at 21:09
• I know everything. :-) – Asaf Karagila Feb 2 '15 at 21:10
• There is one more slogan that I like a lot, attributed to Hrushovski in some writings of van den Dries: "model theory is the geography of tame mathematics" – Darío G May 16 '17 at 14:59
• @Wore: That's probably the closest slogan to the truth. – Kyle May 16 '17 at 20:46