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?
To be clear, we do not have one single system. Branches of math are built on axioms, and different branches include different axioms. Some branches are simple enough that we can prove consistency. But what if you find an inconsistent one? Then you remove one of the axioms that helped demonstrate inconsistency, and then you move on.
It’s a good point. ZFC is a kind of de facto standard, though.
Using the tools of a different, more complex one. Which might itself be inconsistent.
That’s not really a problem, though. There was never a guarantee we can rigorously prove everything.
It is, but the amount of math that relies explicitly on ZFC without being about ZFC is relatively little. Most people don’t think about a particular formalism and the shift to a new one would likely be transparent
And how many people know or care what C does for them, anyway. It’s cool, though.