I've been reading up on some Model Theory lately. My question is simple. What is the purpose of the theory to begin with? How does it enrich mathematics? A-priori, it doesn't seem like we're doing much, and the theory doesn't seem awfully too far from algebra in many instances. For instance, you could interpret the question of $\pi$ or $e$ being algebraic numbers, as a question of definability in model theory. But what's the significance of doing that? Would this be an exercise in futility, since we would have to fall back to algebraic reasonings anyway? Does model theory provide us with new tools to study such questions?
- What's the significance of introducing and coming up with theorems about languages, how does algebra fall short of that?
- What's the significance of studying different models of the same theory? It is tempting to say that they are all the "same" (certainly at least up to some equivalence relation).
Best Answer
While more generality is not always better, we often find it easier to prove a more general result. For instance, although one can prove the unsolvability of the quintic without invoking Galois theory explicitly, that's surely the wrong way to do it: the right way is to develop Galois theory along the way, since it's the natural narrative the problem lives in.
Similarly, model theory is one way of moving from specific settings (e.g. the theory of groups) to more general ones (arbitrary theories). It is certainly not the only way - category theory is a wildly different (and in terms of applications, more successful) example - but it is one way, and it should be no more surprising that it has applications than that any generalization does. In particular, your objections ("But what's the significance of doing that? Would this be an exercise in futility, since we would have to fall back to algebraic reasonings anyway?") could be made to any approach to generalizing anything.
So what are some applications of model theory?
Well, let me begin by addressing an error in the OP and the comments. Theories define classes of structures - e.g. the group axioms constitute a theory, and a model of the group axioms is exactly . . . a group! There are non-isomorphic groups, as I'm sure you're aware. Theories do not in general determine structures up to isomorphism. A version of this (even for complete theories!) can be made precise and proved via the compactness theorem, so your claim that all models of a theory $T$ are "the same" is completely false. There is in general no good sense in which any two models of a theory $T$ are equivalent. (There is an equivalence relation coming from model theory, elementary equivalence, which applies when two structures satisfy all the same first-order sentences - or equivalently each satisfy some complete theory $S$ - but this only makes sense in the context of complete theories, not theories in general.)
Interestingly, compactness turns out to be a useful tool for developing applications! A good example is the Ax-Kochen theorem. Another application is the existence of the hyperreals, a kind of ordered field which allows calculus with infinitesimals to be formally developed; incidentally, compactness here is specifically applied to a complete theory, so it's even valuable to move between two elementarily equivalent structures! And for a much more advanced example, consider the role of compactness for proving transfer principles in motivic integration.
You also dismiss the use of definable elements and sets. This is a mistake. By analyzing the structure of definable (and definable-with-parameters) sets in a structure, we can prove results about the structure itself. O-minimality has been especially useful in this regard.
Finally, connecting back to Galois theory, model theory provides general approaches to Galois-like theories, and was particularly useful in resolving a number of questions in differential Galois theory.
Let me add one more (and then I'll stop since I think I've made my point). A very logic-y theorem is the (downward) Lowenheim-Skolem theorem; roughly speaking, given a "big" structure we may find a "small" substructure satisfying all the same sentences.
Even this result, which is couched specifically in terms of first-order sentences, has applications! There are several examples in general topology that I'm aware of; see e.g. this paper (but there are many others).