Quote:
Originally Posted by Anthony
The issue isn't precisely infinite digits, the issue is infinite information content. If you limit yourself to real numbers that can be expressed by a finite equation, you wind up with a set that is no larger than the set of whole numbers.
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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. If real numbers could not have infinitely many digits, then your proof would be showing only that C was larger than the largest finite number of digits, which wouldn't be very interesting.
But whole numbers cannot possibly have infinitely many digits. The number 0.111 ... is a rational number but the number ... 111. is not a whole number.