We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced results in many branches of ordinary mathematics if we only work with sets and set-membership relation in our language, or otherwise only rely on set theory. To put it differently: it seems that in order to get results in many branches of mathematics one might not need to be very familiar with set theory at all, let alone being able to translate everything to the language of sets or to heavily rely on set theory.
I'm wondering if there are cases where an open/a difficult problem in other branches of mathematics (e.g., number theory or real analysis) has been solved mostly/only because of the insight that set theory has offered, directly or indirectly (say, through branches that heavily appeal to set theory, such as model theory). Even a historical incident will be helpful: a problem of the sort that was first solved thanks to set theory, but later on more accessible solutions have been found that don't deal much with sets.
Thank you very much!
Best Answer
I suspect you'll get a pretty wide range of answers here. There are lots of examples of questions arising in not-set-theory that have turned out to be independent of ZFC. Here's another example that I'm quite fond of, which has a different flavour, in that set-theoretic methods gave an outright answer to a question isn't obviously about set theory.
Let $X$ be a Polish space, and let $B_1(X)$ be the space of (real-valued) Baire class $1$ functions on $X$; that is, functions which can be obtained as the pointwise limit of a sequence of continuous functions. Give $B_1(X)$ the topology of pointwise convergence. Todorcevic proved that every compact subspace of $B_1(X)$ contains a dense metrizable subspace, answering a question that had been raised in functional analysis. His proof uses set theory in a very deep way. As far as I know, no one has found a proof that doesn't involve heavy set-theoretic machinery.
Since your question mentioned model theory, let me also mention Hrushovski's proof of the relative Mordell-Lang conjecture in positive characteristic. His proof used model theory to solve a question arising from number theory. The way that model theory is used in the proof isn't especially set-theoretic, but much of the machinery he used originated in a part of model theory (Shelah's classification theory) that does have strong interactions with set theory.