[jacobolus]

Set theory was developed to make analysis “rigorous” by the standards of 19th century mathematicians, who were probing the edge cases of the more intuitive (or maybe handwavey) assumptions from the 17th–18th century. Here’s Rudin to explain: https://www.youtube.com/watch?v=hBcWRZMP6xs

In the 20th century it became trendy to (a) try to base everything on set theory, (b) write mathematics in a very dry and formal style, taking inspiration from the Bourbaki project.

My impression is that undergraduates are taught in the language of sets partly because it is serviceable for describing the subjects they are trying to learn (esp. analysis) but also more importantly because mid-20th century mathematicians who set up the curriculum we are still using wanted to thoroughly teach (indoctrinate?) undergraduates the trendy style of the time.

Personally I think it has been a mixed bag; the style is a big turn-off to many people, and ends up chasing people out of mathematics who could otherwise make valuable contributions. The people who remain seem to mostly like it okay though. YMMV.

[spekcular]

I have some problems with this comment.

First, why the scare quotes around "rigorous"? Set theory literally did make the foundations of analysis more rigorous.

Second of all, we don't use set theory because it's "trendy" or because we want to "indoctrinate" undergraduates, we use it because it's the best known foundation for mathematics. If there was something less complicated or annoying that worked, we'd use that instead. But as far as we know, there isn't, and there are good reasons to believe we won't find something better.

[jrajav]

Your observations strike a serious chord with me, because the first class I took where a professor wielded set theory like a ruler to the back of students' hands was exactly the point I realized that mathematics was not my calling.

But I've come to understand that I was not realizing that I didn't like mathematics or thought it was too hard, but instead that if I made that my life's work, I'd be working with many more people like that, rather than the people I wanted to - bright and creative, yet kind and humble.

I don't think set theory or any other set of tools is the problem behind potential academics getting turned off. It's the people, and the culture of glorified monkhood.