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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).

namsap
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    Some metric topology might be of use if you're heading to descriptive set theory, and also in the high end where the latter meets large cardinals. You might want to check this question: http://math.stackexchange.com/questions/24854/what-are-the-issues-in-modern-set-theory – Pedro Sánchez Terraf Feb 15 '15 at 14:25
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    It's a bad sad that someone coming online to ask how to approach modern set theory, the answer saying "The hell with set theory, study category theory and topos theory" gets more votes than the answer full of references to actual set theory. – Asaf Karagila Feb 16 '15 at 23:44

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Set theory has numerous applications to several branches of mathematics (Unfortunately, I am a newbie in this field so I do not know most of them). But since you mentioned real analysis and measure theory, let me give some examples from this field that I find really interesting. Some of these may be already known to you but I will give some references (hopefully new to you).

(1) Banach measure problem asks if one could extend the Lebesgue measure on the unit interval $[0, 1]$ to a measure defined on all subsets of $[0, 1]$. By Vitali, we know such an extension cannot be translation invariant but what if we do not require it. This problem is intimately connected to large cardinals (what are now called measurable cardinals) and there are many amazing results and open problems here. When you learn forcing you can read about all of this here - Note the many nice open problems in the appendices. Maybe also check Fremlin's problem list here.

(2) There are many difficult questions about Lebesgue measure on Euclidean spaces whose solutions require forcing - An example is here. Many of these are still wide open. See for example problem GC here. Maybe, also this Erdos paper where the problems have a wonderful blend of geometry, combinatorics, set theory and measure theory.

(3) In a different direction, there are many interesting unresolved problems about sets of uniqueness for trigonometric series. Cantor was originally working on these problems when he invented set theory - Here's a nice talk by Walter Rudin about this. Cohen wrote his thesis on this subject under Zygmund. Kechris has a survey paper on this area.

(4) There is also the field of cardinal invariants with a whole body of results. See for example here and here.

It is nice to have interesting motivating problems (subjective) during your graduate work. It is especially great if these questions were asked by the mathematicians of your previous generations.

hot_queen
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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.

Mathemagician1234
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    I don't understand this answer. Are you saying that in order to learn modern set theory, it is recommended to study category theory? I wouldn't say that it's discouraged in any way, but I don't think I'd say that this is particularly important for set theory. I do agree that understanding the basic notions and concepts is very useful, even if not as an immediate objective, but as a general advice it would be just like I'd advise someone seeking to learn algebraic geometry to learn inner model theory first, and its applications to large cardinals. – Asaf Karagila Feb 16 '15 at 23:20
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    @Asaf Karagila "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?" This is the question I was giving one possible answer to,hoping others would come from other responses. – Mathemagician1234 Feb 17 '15 at 00:00