Our current system? So ZFC?

We move to a different, probably weaker system. Certain mathematicians studying hyper-infinite monstrosities like Banach spaces will be sad. Life will go on, though.

Existing applied math should all work purely in the Von Neumann universe of finite sets. Saying there’s actually an infinite number of points in a line is a kind of luxury; it’s just one that feels right to most mathematicians. A number that’s simply much larger than is worth bothering with can work the same way. In the same spirit, you can probably get rid of most of the Von Neumann universe without breaking practical things.

If you mean actual practical math breaks, I dunno. In any reasonably complex inconsistent formal system, all statements can be proven true. Like 10 = 20. So, can I just grow more fingers somehow?

Edit: Since this is Lemmy, it’s worth pointing out that Godel’s incompleteness theorem has a kind of interchangeability with the fact some questions are undecidable by Turing machine.

Godel’s theorem has a kind of salacious, clickbait quality to it. Laymen will interpret all kinds of things into it. I’ve literally heard “so, math is useless then”. But, if you know algorithms, you know that saying they can’t determine if a loop ends is nowhere close to saying they’re going to stop working, or that they aren’t really good at what they can do.