I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti.
But these all seem to me like pathological, esoteric, ad-hoc examples, that really only matter in foundations, and most non-foundational and applied mathematics wouldn't go anywhere near touching them.
Am I wrong here? If we were to do non-foundational math over naive set theory and just ignore the paradoxes, what problems might we face? Yes, I know that we can technically prove $0=1$ because logic, but I'm looking for more interesting examples, particularly ones that could arise without having to specifically look for them.
Question: Notwithstanding technicalities like explosion, are there any "natural" examples of contradictions arising in non-foundational or applied math due to the paradoxes of naive set theory?
Has anyone ever arrived at a false statement in, say, algebra or number theory, using naive sets?
edit: I'd like to be clear that I'm playing devil's advocate. I'm of course aware that relying on an inconsistent theory is in general a bad idea, but of course not all flawed structures collapse immediately. How far could we go in practice before we ran into problems?
edit: By "non-foundational" I basically mean anything outside of set theory or mathematical logic. If the question of a theory's consistency comes up at all (this thought experiment notwithstanding), then it's probably "foundational". But it's of course fuzzy.