Quote:
Originally Posted by whswhs
I don't think that's relevant to the diagonal proof. The diagonal proof turns precisely on constructing a real number whose first digit differs from the first digit of the first rational (or algebraic) number in a list, whose second digit differs from the second digit of the second rational number in the list, and so on. That results in a number with infinitely many digits that is not in the (by hypothesis) complete list of rational (or algebraic) numbers, but that IS a real number.
|
I am aware of that, but it turns out that the resulting number is not in the set of numbers that can be defined by finite algorithms (with finite inputs), because it's provably possible to map that set of numbers to the integers -- just pick an encoding and the integer map is the encoded form of the algorithm.