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This is kind of a big squooshy question (or series of questions), which I will try to cast in a more precise form. Apologies if I don't succeed.

Context: I'm an amateur set theory/category theory enthusiast with special interest in stratified set theories. My knowledge is substantial in odd spots, but not very well rounded. Because of the ubiquity of talk of models in the literature, it seemed like a thing I should learn, but I'm having a difficult time because I feel like I don't grasp certain informal aspects of "model theory in practice", even though I technically understand about the first half of David Marker's "Model Theory: An Introduction", for instance.

Questions: What are the big questions one wants to answer with models that would either be unapproachable or extremely painful without it? I gather questions of consistency are important ones, but are there other main ones? When does one want to know about the second order properties of elementarily equivalent models, and why? Are there even general answers to these questions, or am I asking things I would have to ask, say, another set theorist about to get anything informative?

I hope this isn't too poor a question; I'm flailing a bit to articulate beyond "it's cool, but WHY???"

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In a very real sense, model theory is something that is really good at defining "dimension" in kind of unfamiliar situations. –  Alex Youcis Nov 11 '13 at 12:37
    
@AlexYoucis: Can you elaborate on that a little? Sounds interesting. –  Malice Vidrine Nov 11 '13 at 16:54
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I should preface this by saying that I am no logician--I just have friends that are. As I have had them explain to me many times, the idea of Morley rank is a fundamental and pervasive notion in Model theory which allows one to extend the notion of dimension to some, somewhat unexpected cases. Also, as I understand it, model theory is really good at talking about categoricity (e.g. Morley's categoricity theorem), which roughly happens when there is some kind of "isomorphism invariant" like dimension hiding somewhere. –  Alex Youcis Nov 12 '13 at 11:42

2 Answers 2

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That's a really interesting question on which we could talk for days, but I hope that in this answer I could tell enough stuff to convince you of the importance of model theory.

The great importance of model theory come from the facts that lots of mathematical structures can be described as models of a first-order theory. One of the more useful things model theory do to us is to translate property of structures into properties of theories (i.e. sets of formulas), which are usually easier to prove to the discrete nature of theories. Using model theory we can prove a lot of things about mathematical structures in a very easy way considering structures as models of some theory.

Here's an example: every graph $G$, even an infinite one, can $k$-colored iff every finite sub-graph can be $k$-colered. This theorem is clearly trivial for finite graphs, nonetheless the only proof I know of this fact uses the compactness theorem.

Lot of algebraic geometry (if not all) over an algebraically closed field of characteristic $0$ can be reduced to the study of algebraic geometry over $\mathbb C$. That's because $\mathbb C$ is a $\mathfrak c$-categorical model of the theory of algebraically closed field of characteristic $0$, and so every first-order property is true for every algebraically closed field of characteristic $0$ iff is true for $\mathbb C$.

Model theory give a lot of techniques for build up models. This is extremely helpful to prove existence of structures of given kind, for instance algebraic ones, that can be described as model of first order theory. On the other end this techniques can also be used in order to prove that some result don't generally hold, by creating counterexamples.

As I said in the beginning there are more stuff could be said about this but I hope this is sufficient to explain way model theory is really important, especially for application in mathematics.

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One advantage is that model theory clarifies the informal distinction between syntactic and semantic aspects of modern mathematics, making it possible to give precise solutions to problems that remained intractable before the advent of model theory. My favorite one (and I am sure some editors will disagree :-) ) is Abraham Robinson's solution of the 300-year-old problem of the consistency of infinitesimals. In the framework of model theory one can give precise meaning to terms such as non-standard models of analysis and successfully formalize centuries of work by the pioneers of analysis who thought of the derivative as a ratio of infinitesimals and of the integral as an infinite sum.

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