On 30 Apr., 16:23, Dan <dan.ms.ch...@gmail.com> wrote: > > 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. > > Let L(n,m) be the m'th decimal of the n'th member of your list of real > numbers . > I define A(m) , the m'th decimal of my number , to be > 2 if L(m,m) not equals 2 , > or 8 is L(m,m) equals 2 . > > Starting from the the fact that a formula for each decimal of every > number of the list was supplied , I supplied a formula to calculate > each decimal of my anti-diagonal . My anti-diagonal is well defined, > and provably not in the list .I didn't write down any digits .
Read what I wrote: 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.
You did not write down any digits. Therefore your example fails. Of course we can define a list like
0.0 0.1 0.11 0.111 ...
and a replacement rule like 0 --> 1. Irrational diagonals can only be created by such rules.
Nevertheless for every rationals-complete list we have:
Forall n: SUM_1^n d_n*10^-n is in the list. Can we abbreviate this by SUM_1^oo d_n*10^-n? Or means "forall n" not for all n?