Elaborating, the hypothesis that Cantor disproves is "The real numbers are countable -- that is to say, the real numbers can be put into one-to-one correspondence with the natural numbers".
You never have to use the axiom of choice, because the hypothesis tells you there is a one-to-one function between the reals and the naturals. You can then order the reals in the order suggested by their image in the naturals: f(0), f(1), f(2), ...
You never have to use the axiom of choice, because the hypothesis tells you there is a one-to-one function between the reals and the naturals. You can then order the reals in the order suggested by their image in the naturals: f(0), f(1), f(2), ...