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?
Similar things have already happened.
Netwon’s laws were rock solid, until we tried to explain really fast or really small things with them. Then we needed Einsteins corrections. Incidentally, we still use Newtons versions for almost everything, because Einstein’s corrections are usually a rounding error.
So if we find a huge flaw, we will immediately start using the correction where it matters, and keep using the old flawed stuff where we’re sure it doesn’t matter.
To be clear: we only continue using the old flawed stuff if it is simpler. If we found one of the formulae we use to describe some system is actually over complicating things whilst also being incorrect, we’d obviously switch in all cases