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?
We know from Gödel’s incompleteness theorem that if ZFC is consistent, then it is incomplete. We don’t consider incompleteness to be a problem any more, so no biggie there. We generally assume and believe that ZFC is consistent and therefore incomplete, a perfectly fine way for it to be. But we can’t prove its consistency, so there are understandably some lingering doubts.
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.