Algorithms work fine on countable subsets of real numbers, as long as the set you use is discrete at some finite resolution. You only run into trouble if you ha

Algorithms work fine on countable subsets of real numbers, as long as the set you use is discrete at some finite resolution. You only run into trouble if you have no lower bound for separation. It's the same problem even if you have only a countable number of numbers, but you have no lower bound for the smallest positive number or upper bound for the largest number. Even integers are non computable in finite time, one one of them might be arbitrarily large.