In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p. > > It is possible to construct the complete binary tree from the numbers > in Q. That means, it is possible to construct all possible digit > sequences from the numbers in Q. It is also possible to construct the > complete binary tree from the set Tp that consists of all termintaing > rationals which are appended by a tail consisting of p. Therefore it > is impossible to distinguish p from the binary tree. But the binary > tree has been errected by using a countable set of paths only.
That ignores the fact that the result of such building automatically contains paths not required in the construction itself.
> Therefore it is impossible to distinguish, by means of digits, any > real number from a countable set of paths.
Not in ZF. > > Conclusion: Cantor's diagonal argument is contradicted.
Wrong outside of WM's world of MathUnrealism. > > Comment: This is not surprising, because it is impossible to identify > any irrational number by means of its decimal representation, but just > this is claimed by the diagonal argument.
Not at all. The diagonal argument, as applied to infinitely long strings of decimal digits, only says that any listing of such strings is incomplete. Which it is.