On 29 Apr., 22:22, Virgil <vir...@ligriv.com> wrote: > In article > <b531ab82-855e-4e8a-b8fe-7ba1a1a07...@b3g2000vbo.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 28 Apr., 23:22, Dan <dan.ms.ch...@gmail.com> wrote: > > > > Give me a list of reals and I shall give you a number not on the list. > > > No, you will fail. > > The only proof that he must fail wold be by producing a list of numbers > which contains all real numbers.
A real number can be listed by a terminating decimal, by a periodic decimal, or by a formula supplying each of its decimals.
It is not possible to list irrationals by writing their decimals. Therefore it is not possible to create irrational diagonals of Cantor- lists by writing their decimals.
For every rationals-complete list, we have:
forall n in |N : (d_1, ..., d_n) are the first n digits of infinitely many numbers in the list.