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Is reading Jech's text on Set Theory too little, just enough, or overkill to prepare oneself to do independent research in set theory? This would be my first attempt at doing independent research after taking 5 graduate level math courses and doing well. If I think I'm interested in Set Theory, and want to get a feel for research, is this the way to proceed?

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The best (only?) way to prepare oneself for research is to to enroll in a graduate program and follow the advice of your advisor. In any case, this question is off-topic for MO and I have voted to close. –  Andy Putman Jun 25 '13 at 20:04
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It sounds likely that OP hasn't yet approached a potential thesis advisor (or hasn't yet been accepted as a student of such), in fact might be an undergraduate, and just wants an idea if Jech will bring up him/her to speed with cutting-edge problems. I think that's more or less a reasonable thing to ask. –  Todd Trimble Jun 25 '13 at 20:15
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Andy, could you explain why not? Perhaps it should be CW? I think that it is a question that could easily be asked by a graduate student at math tea, and it can only be sensibly answered by someone with lots of familiarity with the text and with what it takes to undertake research in set theory. –  Joel David Hamkins Jun 25 '13 at 20:55
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Well, an answer of "no" could be a fine answer as well! But I don't think it's entirely clear cut, just on general principle. One can imagine a very clever undergraduate reading, say, Diestel's Graph Theory, and then solving thesis-worthy problems in graph theory through some general knowledge of what has been done and by dint of sheer cleverness -- this type of thing does happen. In other words, I think it really depends on the type of mathematics -- and this is where informed professionals such as Joel can provide valuable and nuanced advice. –  Todd Trimble Jun 25 '13 at 21:25
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There's a false premise behind the question, which is that the way to prepare to do research is to read a book. Reading is part of it, but what you really need is to develop a feeling for what problems are (1) unsolved, (2) important, and (3) tractable. It's very hard to develop this feeling on your own (though a few exceptional people can do it). Reading a book is certainly not sufficient. Conversely, it is not necessary either; for any particular problem you end up solving, much of Jech will probably be irrelevant. That said, if you don't have an advisor right now, reading Jech can't hurt. –  Timothy Chow Jun 27 '13 at 4:15

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I believe that Jech's book is a solid part of any graduate student's preparation for independent research in set theory. He covers most or even all of the main topics of set-theoretic research, and he does so at quite a high level, including some extremely advanced material. For further study of large cardinals, however, the book should probably be supplemented by Kanamori's book The Higher Infinite (see my review), and for learning forcing, I always encourage my graduate students to read both Jech and also Kunen's book, as well as some others, especially Bell's book on forcing via Boolean-valued models (and my own article on the Boolean ultrapower), and to play all these texts off of one another, as each has some strengths the others lack.

Jech's book is extremely thorough, and I suppose that if you mastered every last bit of it, then indeed I think it would position you for independent research in set theory. But of course, the more typical pattern is to read at first only the easier parts of it, while also learning from other books, and gradually bring oneself to the research level that way.

During and after your study of Jech, you will need someone to guide you to research topics and problems, to suggest problems or areas that might be fruitful for your independent work or which may interest you.

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But Joel, Jech's is only thorough on introductory material, don't you think? If one wants to learn forcing axioms, or pcf theory, or determinacy, or reflection principles, or just about any current topic, Jech's book only provides a cursory presentation. –  Andres Caicedo Jun 25 '13 at 20:28
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Andres, I would add that you and I probably don't agree 100% on what counts as a "current" topic in set theory, since I find it likely that I may be more of a generalist than you, and to be a generalist takes a different kind of preparation. –  Joel David Hamkins Jun 25 '13 at 20:37
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(I have a plan. Maybe in a year. I'll let you know.) –  Andres Caicedo Jun 25 '13 at 21:08
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My own path to set theory involved reading through the green version of Jech's book back in the late 80s. No one at my undergrad did set theory, but I ran across Jech's book in the library and I was hooked. –  Todd Eisworth Jun 26 '13 at 14:36
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Of course, things that took me weeks to figure out on my own were clarified in a few minutes of discussion with Andreas once I got into grad school... –  Todd Eisworth Jun 26 '13 at 14:58

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