This is coming from the opposite extreme. In constructive mathematics, abandoning Countable Choice may allow: - The complex numbers to not be closed under squa

This is coming from the opposite extreme. In constructive mathematics, abandoning Countable Choice may allow: - The complex numbers to not be closed under square roots. - Space-filling curves to no longer exist. And subtle complications around: - The definition of a real number. The Dedekind and Cauchy real numbers may become non-isomorphic. As it stands, the following has not been validated in constructive mathematics without countable choice: - That the Dedekind real numbers are uncountable. - That if a function over the Dedekind reals has derivative 0 everywhere, then it's constant. On the other hand, the Approximate Intermediate Value Theorem has been proven, but with a weird proof. See here [1] for more. [1] - https://ncatlab.org/nlab/show/countable+choice https://ncatlab.org/nlab/show/countable+choice