TITLE: Which other fields of mathematics are relevant to modern set theory?
QUESTION [7 upvotes]: I am currently studying mathematics as an undergrad. Over the last year I discovered mathematical logic and set theory and took some courses in these subjects. Especially set theory is really appealing to me, so naturally I am interested in deepening my knowledge of it. I think set theory is the most interesting part of mathematics I've seen so far. Currently, my aim is to become a professional set theorist. To do so I obviously have to learn more about set theory and take every chance to attend set theory courses. As an undergraduate student, I have (and of course want) to cover different areas of mathematics and not only set theory.
So I wondered - which other areas of mathematics are important in modern set theory or have an influence there? What other areas should one study, if he wants to become professional in set theory?
I am happy to get any general advice on which areas to study, but would also like to hear especially which parts of these areas are important (for example measure theory in analysis).

REPLY [2 votes]: One of the critical topics in the foundations of mathematics is establishing a firm set-theoretic foundation for category theory. Saunders MacLane, the co-founder of category theory, was quite concerned with this issue and attempted several possible solutions, the most sophisticated being topos theory, which basically replaces axiomatic set theory with a specialized theory of categories called topoi, which behave like sets. There are some who believe this is a purely metaphysical quandry, but I couldn't disagree more.Using category theory fluently to express results as many mathematicians do while avoiding this question like a land mine is,to me, tantamount to what the 18th century mathematicians and physicists did during the non-rigorous phase of the history of calculus. It is a very important question of rigor and one that I believe may lead to important results in the foundations of mathematics.Also,frankly-we shouldn't be comfortable using category theory as a "black box" in which we express our theories, any more then mathematicians should have been comfortable with 19th century Italian algebraic geometry-which was basically a game of pictures and metaphysics until Oscar Zariski and other algebracists established firm foundations for it in the 1930's and 1940's.