Even programs with infinite descriptions are only countably infinite, but the reals are uncountable.
This construction by Cauchy sequences looks innocent, but it's the diagonalization argument in disguise. (Start with all the rationals, make sequences out of them, make one that picks one from all, sort them into equivalence classes, and try to map them back to the rationals, notice that you will end up with more equivalence classes.)
The trick is basically that between every rational you can fit an infinite number of irrationals (using the rationals via these sequences). And exactly in this way these are "programs" -- like diagonalization itself. The fact that we can't give programs for most of them is because they are non-computable. (And it's the definition, the indirect proof is above via the cardinalities.)
This construction by Cauchy sequences looks innocent, but it's the diagonalization argument in disguise. (Start with all the rationals, make sequences out of them, make one that picks one from all, sort them into equivalence classes, and try to map them back to the rationals, notice that you will end up with more equivalence classes.)
The trick is basically that between every rational you can fit an infinite number of irrationals (using the rationals via these sequences). And exactly in this way these are "programs" -- like diagonalization itself. The fact that we can't give programs for most of them is because they are non-computable. (And it's the definition, the indirect proof is above via the cardinalities.)
[but it's dangerously late here, so double check my ramblings ... https://math.stackexchange.com/a/1488502 ]