I'm talking about topics such as model theory, proof theory, predicate calculus, Godel's incompleteness, ordinal numbers, large cardinals, etc. How much of these do I need to know if I want to specialize in some other areas of mathematics such as algebra and topology?
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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