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A rather straight to the point soft-question: What kind of background should have somebody who wants to study properly descriptive set theory?

More specifically, how much analysis should she/he know?
Beyond the actual amount of knowledge, how well should this person deal with standard $\epsilon - \delta$ arguments?

Do you have any advice for somebody who wants to embark in this project?

Please, notice that I am completely self-taught, thus I have the feeling that the advice you would give to somebody who can be in some way tutored could be slightly different.

Thanks as always for your time and your feedbacks.

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    $\begingroup$ This will be based on long-ago memories of having studied set theory, but I'd say that if you are comfortable with general topology and measure theory, that should be enough. I imagine it is difficult to acquire a sufficient level of competence in these areas without having gone through the $\epsilon--\delta$ stage of analysis first. When I studied logic, I remember seeing classmates with less background in "regular" math, who were coming from computer science or philosophy, and they had trouble even in the parts that had no direct prerequisites. The reason is that the process of learning ... $\endgroup$ – user204305 Jan 8 '15 at 3:58
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    $\begingroup$ ... abstract algebra and analysis in the normal way equips you with a certain outlook on mathematical questions, and perhaps also better problem-solving ability. So as a practical matter, this is the normal route to acquiring the "mathematical maturity" you need to attack more advanced subjects. I don't know whether it is easy to gain this maturity through other routes, but what I observed was that it was very difficult for others to get it by studying logic and set theory. $\endgroup$ – user204305 Jan 8 '15 at 4:02
  • $\begingroup$ @user204305: Without disagreeing completely, let me point out that I have taught many a freshman set theory and logic courses, and even in the current course which is naive set theory aimed for advance undergrads, we still have a reasonable portion of freshmen. It is possible to gain mathematical maturity with set theory and logic. That being set, descriptive set theory probably requires to be comfortable with topology, a bit of measure theory, and basic set theory. To properly learn the topic, deeper understanding in set theory is probably needed. $\endgroup$ – Asaf Karagila Jan 8 '15 at 7:29
  • $\begingroup$ @AsafKaragila Classmates who had never studied much about groups and rings had a hard time when it came to proving theorems about "structures" (or even just examples of structures), or using Zorn's lemma to prove the completeness theorem. People who had never studied the ordering of the real numbers properly had trouble with proving the countable categoricity of dense unbounded orders. And so on. A certain mathematical outlook pervades all of logic and is actually a prerequisite for it, even though you often don't need the facts of analysis itself. I believe that because of the more... $\endgroup$ – user204305 Jan 8 '15 at 7:45
  • $\begingroup$ ...abstract nature of the objects studied, compared to basic analysis and algebra, it's more difficult to learn this way of thinking through set theory and logic alone. I'm not talking about basic facts on countable sets, etc, which are a part of the background you should get in analysis/algebra. $\endgroup$ – user204305 Jan 8 '15 at 7:49

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