Generally mathematical theorems involving real numbers, do not care about computability (except if you are a finitist https://en.m.wikipedia.org/wiki/Finitism ) ;)

On the other hand my intuition is: flipping a coin (generating next digit of a real number between 0 and 1) and then trasforming it to e.g. an exponential random variable should always take you to the point where you can decide about the ordering in countably finite coin toss, with probability one...