Can you in practice use set theory to discover something new in other branches or math, or does it merely provide a different (and less convenient) way to express things that were already discovered otherwise?
The value of set theory as a foundation comes more from being a widely-agreed upon language that is also powerful enough to express pretty much everything mathematicians can think up, rather than as a tool for making new discoveries. I think it’s worth learning at least at a shallow level for this reason, if you want to learn advanced math.
I’d add that set theory gives you tools (like Zorn’s lemma and transfinite induction) that aren’t particularly exciting themselves, but you do need them to prove results elsewhere (e.g. Tychonoff’s theorem, or that every vector space has a basis).
That said there are some examples of results from formalizing math/ logic being used to prove nontrivial things elsewhere. My favourite example is that the compactness theorem of first order logic can be used to prove the Ax–Grothendieck theorem (which states that injective polynomials from C^n → C^n are bijective). I find this pretty cool.
The value of set theory as a foundation comes more from being a widely-agreed upon language that is also powerful enough to express pretty much everything mathematicians can think up, rather than as a tool for making new discoveries. I think it’s worth learning at least at a shallow level for this reason, if you want to learn advanced math.
I’d add that set theory gives you tools (like Zorn’s lemma and transfinite induction) that aren’t particularly exciting themselves, but you do need them to prove results elsewhere (e.g. Tychonoff’s theorem, or that every vector space has a basis).
That said there are some examples of results from formalizing math/ logic being used to prove nontrivial things elsewhere. My favourite example is that the compactness theorem of first order logic can be used to prove the Ax–Grothendieck theorem (which states that injective polynomials from C^n → C^n are bijective). I find this pretty cool.