But that's nothing special about set theory. I prefer to think that the role of mathematics (at least the best kinds of mathematics) is to correct and extend our intuition, not to "transcend" it. But the kind of powerful experiences you describe were available two thousand years before the invention of set theory, and they're available all over modern math in areas that have nothing to do with set theory.
Not every student would benefit from learning set theory early beyond the universally needed understanding of injective/bijective mappings, but some would. It does depend on personality. It has some relation to cultural things.
"Role of mathematics" implies relatively long run; experience is felt in a very short run.
If you want to extend your intuition into an area nobody understands well, you often need to combine quite weak analogies and formal methods - because you need to do something to get any useful intuition.
There are many branches of s...
Did computer programming make you a clearer, more precise thinker? How about mathematics? If so, what kind? Set theory? Probability theory?
Microeconomics? Poker? English? Civil Engineering? Underwater Basket Weaving? (For adding... depth.)
Anything I missed?
Context: I have a palette of courses to dab onto my university schedule, and I don't know which ones to chose. This much is for certain: I want to come out of university as a problem solving beast. If there are fields of inquiry whose methods easily transfer to other fields, it is those fields that I want to learn in, at least initially.
Rip apart, Less Wrong!