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I have a question that has occupied my mind for some time.

Let's first consider applications of set theory and model theory in mathematics.

Major applications of set theory are in topology, Banach spaces, $C^*$-algebras, functional analysis, dynamical systems, algebra,....

On the other hand, major applications of model theory are in number theory, arithmetic and algebraic geometry, algebra,...

What is now interesting to me is that when set theory is applied at some part, model theory has few to say about that part, and vice versa, when model theory is applied heavily in some part of mathematics, then set theory appears rare there.

Even though both fields have applications in algebra, when we restrict ourselves to sub-branches of algebra, again the above obsession is true.

I am wondering if my observation is correct and if so is there any reason for it? Or should we just wait for more convergence in the future?

Remark 1. On the other hand surprisingly there is deep connections between set theory and model theory.

Remark 2. One answer to the question might be, set theory in the sense of model theory is wild (and not tame), but this is not quite convincing to me.

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  • $\begingroup$ Related to your second remark: I would say model theory is mostly about first order statements, whereas non elementary set theory deals with higher order statements. $\endgroup$
    – nombre
    Commented Jul 16, 2020 at 13:15
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    $\begingroup$ I'm not sure that your observation is correct. For example, in addition to the applications of set theory to $C^*$-algebras, there are also many applications of model theory to the field: arxiv.org/pdf/1602.08072.pdf $\endgroup$
    – Haim
    Commented Jul 16, 2020 at 15:59
  • $\begingroup$ @Haim Yes I know that, even for example I can say Shelah's proof of independence of Ax-Kochen isomorphism theorem from CH vs. the model theoretic works on the subject. But if for example you look at the paper you sent, I think you agree that the subjects of the applications are different from what set theory has, also I think the applications stated are not as deep as those given by set theory. $\endgroup$
    – Mohammad Golshani
    Commented Jul 16, 2020 at 16:54
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    $\begingroup$ Dynamics is another example of an area where both set theory and model theory have many applications (again, I don't know which of them satisfy your criterion for being deep), one recent example is this paper by Hrushovski, that makes an interesting connection to the Kechris-Pestov-Todorcevic correspondence: arxiv.org/pdf/1911.01129.pdf $\endgroup$
    – Haim
    Commented Jul 16, 2020 at 19:14
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    $\begingroup$ @TimothyChow It seems to me like an almost empirical fact that in order for a non-set theoretic statement to be susceptible to set theoretic methods, it should either be a statement involving subsets of Polish spaces with a very simple (Borel/analytic) definition (in which case methods from DST can be used) or it shouldn't be absolute (so forcing, large cardinals and inner models might be useful). It seems that non-set theoretic statements that satisfy either of the above requirements are harder to come by, whereas such requirements don't exist in the case of model theoretic applications. $\endgroup$
    – Haim
    Commented Jul 17, 2020 at 3:29

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