I know enough amount of set theory and logic to study grad-level math. However, I don't know more advanced set theory and logic, such as the ones on Kunen's or Shoenfield's texts. Although it's good to learn such advanced topics, I may not have enough time to do so, since I will probably put my emphasis on algebra, analysis or geometry as well as mathematical physics.

Although how much set theory and logic I should learn is totally up to what sort of analysis, algebra or geometry I will study, I want to know how much of them typical competitive algebraists/analysts/geometers know. If you are one of them, how much do you know?

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    $\begingroup$ Advanced set theory such as forcing or constructible universe does not need for ordinary mathematicians, so (as I know) almost ordinary mathematicians doesn't know about them. I saw a person who is intelligent for analytic number theory confuses the undecidability of continuum hypothesis. $\endgroup$ – Hanul Jeon Dec 30 '14 at 3:01
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    $\begingroup$ I suppose I'm an algebraist, and I only know a little logic, which I happened to learn from studying computer science, and this is probably more than half the algebraists out there. I think most mathematicians have a passing familiarity with logic and set theory, but don't have a serious understanding of the subject (e.g., most probably don't have a clear understanding Godel's incompleteness theorems). I also don't think this is a serious hinderance in practice. $\endgroup$ – Kimball Dec 30 '14 at 3:50
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    $\begingroup$ I am a set theorist, and I can tell you that for the majority of fields, set theory and logic are not necessary. As Kimball wrote. However, knowing set theory and model theory can be an additional tool in your toolbox and it might be one that gives you edge over "the competition". The fields you mentioned are a bit broad from this aspect, but they have been known to interact with set theory and model theory in some cases. Not to mention, that god forbid, you end up being a set theorist, or a model theorist. ;-) $\endgroup$ – Asaf Karagila Dec 30 '14 at 3:55
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    $\begingroup$ That actually means you're unlikely to ever do it. You tend to gravitate towards things you are interested in, and those will usually be the things you "almost know". If you come to algebra knowing set theory, you'll be easily drawn to more model-theoretic aspects; whereas if you come without any actual knowledge of logic, you'll easily gravitate towards other parts of algebra. It's fine, just know that this is a likely trajectory of your mathematical future. $\endgroup$ – Asaf Karagila Dec 30 '14 at 4:18
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    $\begingroup$ Algebraic geometry from the perspective from model theory is a large topic right now. Is it necessary to do model theory to understand algebraic geometry? No. Might it help you gain perspective? Yes. However, the amount of time you would have to invest in model theory to understand model theories application to this area might not be worth it. $\endgroup$ – Kyle Gannon Dec 30 '14 at 7:47

I would say at least what is contained in Halmos' Naive set theory. The preface reads:

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set­ theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds."

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    $\begingroup$ Is this book about set theory version that admit paradoxes (contradictions) e.g. Russel Paradox? $\endgroup$ – Trismegistos Jan 8 '15 at 15:11
  • $\begingroup$ The title says "naive", but the set theory dealt in the book is just an axiomatic set theory which doesn't lead to such a paradox. The author used the term "naive" to mention the gentleness of the treatment of set theory in the text. $\endgroup$ – Math.StackExchange Jan 13 '15 at 13:14

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