TITLE: Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?
QUESTION [26 upvotes]: 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!

REPLY [11 votes]: Dehornoy order in braid theory was first discovered using large cardinal axioms. For a while it was an open problem if large cardinal axioms were actually required, although since then elementary approaches have been discovered.