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Does being bad at elementary set theory, mean someone will be bad at proofs in upper level courses like real analysis, topology, algebra? I can't write Cantor's diagonalization argument as a non-proof by contradiction because I don't understand it well enough. I couldn't prove that the cardinality of Cartesian products of set $M$ and $N$ is $M\times N$ unless I'm spoonfed the answer. I can't understand what a dense subset is. Is analysis, algebra, geometry, not for me? I don't know why we use Cartesian products to prove that the countable union of countable sets is countable. P-adic numbers don't make sense to me. I couldn't tell you why a subsequence is either strictly increasing or decreasing. I don't understand transcendental vs algebraic numbers. Well-ordering principle I can use as a formula but I don't really understand WOP.

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    You don't know anything until you learn it. If you want to learn it, then keep working on it. If you don't, then quit. – MPW Jun 22 '17 at 13:45
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    Don't be discouraged. Everybody has strengths and weaknesses, and transfinite numbers are very unintuitive. Work through the bits that are hard to get, it gets easier over time. – lulu Jun 22 '17 at 13:47
  • Elementary set theory is rather simple. Maybe you need the right book to get a hand on it. – Wuestenfux Jun 22 '17 at 14:07

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At some point, every one learns what it is to struggle. About most concepts, I have nothing intelligent to say, assuming I even recall the definitions. I cannot solve most problems that are out there in mathematics, and I expect that most people in research have similar experience. We need to see examples of how proofs go in order to structure our ideas, and spend a great deal of time to become familiar with concepts.

I know that if I keep at it, stay struggling long enough, think about things long enough and in enough different ways, eventually, I'll get it. And then the thing that I figure out will become an experience which may help me with a different problem.

Eventually you'll get it too. Welcome to mathematics :)

A. Thomas Yerger
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Leibniz did just fine with analysis, and even algebra and geometry. I assure you that not only did he not know a thing about set theory but he actually opposed the actual existence of infinite sets taken for granted in modern set theory. It would have been gibberish to him as it is to you.

Mikhail Katz
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    Leibniz didn't struggle with set theory though. The question is not about "do I have to understand set theory well to be good at other things in maths" but rather more in flavor of "does being unable to grasp set theory show that I don't have potential to pick up maths". – Wojowu Jun 22 '17 at 14:04
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    That's precisely my point. The skills necessary to learn set theory may be quite disjoint from the skills necessary to learn analysis. – Mikhail Katz Jun 22 '17 at 14:42
  • If you are bad at set theory ,point set topology will be difficult because there is plenty of set theory used. The introductory chapters in many books start with set theory,relations and functions? For algebra l don’t know,probably far less than point set topology For Analysis knowing the basics could help –  Sep 08 '21 at 20:16