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?
Ah, thank you for that clarification!
So, for example, having an axiom that says you can’t have a set that includes sets that don’t include themselves makes it incomplete, but we don’t consider that a big deal because it is more useful for it to be coherent than complete.
By coherent I assume you mean consistent. Yes an inconsistent theory isn’t considered useful in mathematics. In philosophical logic there’s an idea of “paraconsistency” that means something like “inconsistent but only slightly” but I think it’s not used much in math.
Russell’s fix to Frege’s inconsistent system was quite complicated, much more than just adding an axiom disallowing certain types of sets. ZFC handles it differently too, by saying you can only create new sets by following certain rules designed to keep things consistent. Frege’s system let you do whatever you wanted and it went sideways quickly, as Russel found.