Set theory also helps one see mathematics as a whole and see how different areas connect to each other.
For what it's worth, I strongly disagree. For a new student too much emphasis on foundations can be a major mental block when getting used to a new idea and especially a new circle of ideas. Set theory is used very informally in most of mathematics, as a notation. To learn more than this notation is mostly unnecessary for pure math, completely unnecessary for applications of math to other areas.
Well, the difference of mathematics from natural sciences that you need not only to build a good model of something, but also to describe it using a limited set of axioms (using proofs from there on).
For some people set theory is the area of mathematics which quickly reaches proofs that are accessible to our reasoning, but transcend our intuition. Sometimes even the notions used are easy to define formally but put strain on your imagination. For people inclined to mathematics this can be a powerful experience.
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.
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!