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Is set theory dying?

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A lot of mathematical research is about what's in "fashion" at the moment - what are people currently publishing papers on? What are people talking about in conferences? etc.

For whatever reason (not my field), set theory has gone out of fashion. I think lots of people are into model theory now, which is somewhat adjacent as part of logic.

Edit: based on other comments, set theory is not out of fashion (I just assumed based on what OP initially said; my bad)
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Which departments do you see set theory drying up at? Within logic, the narrative right now is that set theory is flourishing and there’s a new generation of successful researchers proving exciting theorems and doing fairly well on the job market.
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How "healthy" has set theory ever been? When I look up the famous set theorists (Cantor, Zermelo, ...) they have very few students. Meanwhile, Hilbert at roughly the same time has tons of students. Some students (e.g. Weyl) become famous and had tons of students themselves (in algebra/geometry areas).

In almost every"hot" area I know I can think of some advisor(s) who put tons of students into the system and some of those students have started making more students.

Maybe there is a prolific advisor in set theory I am unaware of?
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Defining "dying" in terms of a "research area" makes more sense. This is to be expected over time when progress, or expectations of progress, slows. Essentially becoming a mature framework no longer seeing major developments. This can change if, or when, developments occur that reignites progress in set theory. But presently it's mostly more of a tool set than an invention in progress. In that sense it merely matured rather than dying, and no longer requires the level of research needed to mature it.
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I think set theory is so advanced since so many people have worked on it that the open questions are about as hard as anything, and it’s likely easier to get funding for problems in other areas just as difficult from NSF and others that those talented enough to tackle such problems work on other things. Add in a little less interest compared to other areas, and you have what you see. Just a thought.
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"Pure" Set theory is fundamentals research which is in a perpetual near-death state but also never seems to fully die. The problem is that the number of theorems provable with just sets is very small, or the theorems are very hard, or they aren't that useful to broader mathematics.

This is not to say that it's useless, but one should remember that set-theory was never that popular or big of a field in the first place.

One relatively recent change though is the "mainstream" advent of alternative foundations for mathematics which (imo) are a lot prettier and richer in structure and therefore theory. This means that the foundational status that Set theory has enjoyed since Bourbaki is starting to wane, which means that people that would originally have done foundational research in set-theory may wander off to those.
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I think my dad has a phd in set theory... hasnt explained it much to me in the past 30 years
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No branch of mathematics ever dies. It might fade out of popularity but that’s all.
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Today, Category Theory is attracting all the talents, who would have otherwise gone into Set Theory 40 years ago.
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Let me make a somewhat disparaging comment about mathematicians:

I think that most mathematicians (even research mathematicians) have very little interest in the metaphysical underpinnings of their discipline and quite a few hold onto some unrefined platonism (nothing wrong with platonism), otherwise we would see more people engage with category theory or set theory. Of course, one can do both of these without thinking about these philosophical questions, but at least some set theorists like Woodin seems to engage with set theory because of the need to paint a certain picture of the real subject matter of mathematics.

Edit: A lot of mathematicians seemed to be offended by the phrase "real subject matter". As I have written below, "real" does not mean better or more valuable but more basic and potentially revealing what mathematics is at its core. "Real" might mean something  like more basic and capable of being a basis to which other mathematical objects might be reduced to.

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And to what extent is the lack of young talent due to  poorly written literature? As for introductory textbooks Enderton and Jech come to mind, but the costs of these books is insane for the amount of pages they deliver.

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