From my “watched a YouTube video” understanding of Gödel’s Incompleteness Theorem, a consistent mathematical system cannot prove its own consistency, and any seemingly consistent system could always have a fatal contradiction that invalidates the whole system, and the only way to know would be to find the contradiction.
So if at some point our current system of math gets proven inconsistent, what happens next? Can we tweak just the inconsistent part and have everything else still be valid or would we be forced to rebuild all of math from basic logic?
It wasn’t a mistake. Usually you’re talking about an infinite-dimensional TVS when you say Banach space - as in it’s just Banach, that’s the most you can say about it. I don’t mean R3.
Stuff like the axiom of choice has a way of coming up in functional analysis. Sure, there’s weirder spaces, like from general topology or TVS theory in general, but Banach spaces are an example that are pretty widely used and studied. It seems like going with some pathological object I have to search around for would make the point less clear.
The Banach-Tarski paradox wouldn’t work either without the AoC, but it’s just a specific counterexample, and Banach and Tarski’s careers will be fine since they both died decades ago.