You wouldn't even notice if some proof is wrong because it relies on an inconsistency that's the issue. And that's before you didn't notice because noone builds anything on axioms but instead uses fragile foundations made of intuition, hand-waving, and mass psychology.

Incomplete is fine that's exactly what constructive maths is doing.

https://en.wikipedia.org/wiki/Continuum_hypothesis

Hmm. Frankly speaking I always assumed Mathematicians had more of an idea about infinities I mean why even have indices if you don't have an inductive rule to descr... oh wait never mind.

That said the reals aren't countable yet we have perfectly reasonable ways to deal with them symbolically, even compute with them, represent every single one of^1^ them in finite space, it's just when you want to compare or output them with infinite precision that you might have to wait for eternity. But who needs infinite precision anyway, arbitrary precision is plenty.

^1^ On second thought, after diagonalisation, no we don't. Or we do because there's some magic going on with included transcendental constants that break through that do I look like a numerologist.

(which happens to work similarly to Cartesian coordinates, so that's convenient)

it's more than convenient, it's isomorphism!

I don't understand what you think the problem is. What do you mean infinities can't be differentiated from each other? Infinite cardinals are by definition equivalence classes of sets that can be placed in bijection with one another. You can compare one cardinal to another by showing there exists an injection from a representative set of the first one into a representative for the other. You can show equality by showing there is an injection both ways (Cantor-Schroder-Bernstein theorem) or otherwise producing a bijection explicitly. Infinite ordinals may as well be copies of the natural numbers indexed by infinite cardinals, so of course we can distinguish them too.

O(n^m^) = O(1) for all n,m < infinity.