> Even with infinite amount of time your enumeration function will not generate all of the natural numbers even though the natural number are countable. Huh? Y

> Even with infinite amount of time your enumeration function will not generate all of the natural numbers even though the natural number are countable. Huh? You can trivially write a function to generate all natural numbers: def increment(x): return increment(x + 1) increment(1) How do you propose doing that for the reals? > Where it's difficult is when we talk about numbers that aren't describable in a finite number of terms, i.e. Uncomputable. This is not what uncomputable means. Uncomputable means we cannot calculate a number to any precision, not that the number has an infinite decimal expansion. Pi is not uncomputable. You can’t store its entire decimal expansion in finite space but it is certainly computable - we compute it all the time. However, almost all real numbers are strictly uncomputable. Among many other proofs, this is evident from the simple fact that the set of all possible Turing machines is countable. Since the reals are uncountable, it follows that almost all reals must not be computable by any Turing machine. Likewise, there are many potential calculations involving real numbers for which a computer will output a strictly incorrect - not just imprecise - result because its real solution is uncomputable. This does not happen in the naturals or rationals.