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?
Mathematics is just a language to describe patterns we observe in the world. It really is not fundamentally more different from English or Chinese, it is just more precise so there is less ambiguity as to what is actually being claimed, so if someone makes a logical argument with the mathematics, they cannot use vague buzzwords with unclear meaning disallowing it from it actually being tested.
Mathematics just is a language that forces you to have extreme clarity, but it is still ultimately just a language all the same. Its perfect consistency hardly matters. What matters is that you can describe patterns in the world with it and use it to identify those patterns in a particular context. If the language has some sort of inconsistency that disallows it from being useful in a particular context, then you can just construct a different language that is more useful in that context.
It’s of course, preferable that it is more consistent than not so it is applicable to as many contexts as possible without having to change up the language, but absolute perfect pure consistency is not necessarily either.