As an undergraduate, I noticed none of my courses in analysis, topology, or algebra require the need to know any set theory beyond naive set theory. My question is is this true for even higher mathematics, or will I need to pick up a book on axiomatic ZFC set theory if I try to become a pure mathematician?
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Some background: Modern mathematics, being a very young subject, is still heavily influenced by the historical development in the past, say, 400 years. Thus, the standard textbooks mirror to some degree the evolution of mathematics. Such pillars as calculus and linear algebra existed long before logic and set theory (the latter are only about a century old) you hardly see any use of set theory in these subjects. In fact it took some time to identify the ubiquity of the axiom of choice in analysis, not because its difficult to spot, but simply because considerations of the axiom of choice were rather later, well after the calculus was quite well-established.
Without a doubt, if the entire mathematical body of knowledge (including textbooks) were to be re-written things would look quite different than they are today and would probably include some considerations of axiomatics.
Some techniques of logic and set theory (e.g., the compactness theorem and Zorn's Lemma) are in the standrad toolkit of many mathematicians who do not specialize in logic or set theory. Formally speaking, you don't need to have these in your toolkit as well but they do not require a great deal of time to master and not having them is sometimes a pity.