1. I thought mathematics was moving more towards a category theory based foundation, as opposed to the set theoretical stuff.
2. Isn't it in part because we have over 120 years of results and proofs that are based on set theory, and people aren't just going to throw that out for something newer? Granted, Category Theory has been around in some form since about the 1940's, but it's still newer than set theory.
3. Russell's Paradox hasn't stopped people from using Set Theory to achieve useful results, and the axiomatization of Set Theory and the advent of ZFC is why (or at least part of why) that was possible, no?
1. It's not. The work on "category-theoretic foundations," say HoTT, is (sociologically speaking) a highly niche topic. Most professional mathematicians who work on the foundations of mathematics do so in the framework of set theory.
2. ZFC is fully adequate for all the mathematics 99% of professional mathematicians do (and probably more than adequate in terms of strength). Also, all the mathematics 99% of mathematicians do is insensitive to foundational issues, so if you swapped ZFC for another axiomatization of similar strength, no one would notice.
3. I don't believe ZFC was the first to avoid the paradox, but yes, it does not suffer from Russell's Paradox.
1. I thought mathematics was moving more towards a category theory based foundation, as opposed to the set theoretical stuff.
2. Isn't it in part because we have over 120 years of results and proofs that are based on set theory, and people aren't just going to throw that out for something newer? Granted, Category Theory has been around in some form since about the 1940's, but it's still newer than set theory.
3. Russell's Paradox hasn't stopped people from using Set Theory to achieve useful results, and the axiomatization of Set Theory and the advent of ZFC is why (or at least part of why) that was possible, no?