I know about that, AoC, CH, Gödel, Turing, Cantor, Hilbert and such, I had a course on mathematical logic and one on set theory when I was an undergrad and the

I know about that, AoC, CH, Gödel, Turing, Cantor, Hilbert and such, I had a course on mathematical logic and one on set theory when I was an undergrad and the lectures about Gödel were the high point in both of them, although in set theory there were some high points with the idea of "set of all sets" and some other things. AoC is used indirectly in almost every place of pure mathematics, but I researched numerical methods for PDEs and Stochastic PDEs which is not pure mathematics, maybe someone who is much more intelligent than me will say where you will use AoC directly in this area outside of some theorems of Analysis that are used but well, I've never ever touched this same axiom again in my life and if I chose to spend my life as a researcher I doubt I will ever need it again to work and publish, and yet I was able to get a Ph.d in Applied Mathematics. The theorem that I know uses the AoC that I cited once but not exactly used is Banach–Alaoglu theorem. As sbi put if you go to a department of mathematics that includes mathematicians, statisticians and applied mathematicians chances are that almost no one will know too much of set theory excluding some pure mathematicians, this was true on almost every department that I saw in my entire life. So there are mathematicians whose life are not spent trying to use abstractions everywhere. That was the point.