On 1 Mai, 18:27, Dan <dan.ms.ch...@gmail.com> wrote:
> They have similar formulas,but behave in different ways . > You would be correct in affirming that b_inf = 1/9 is not part of the > list .
Of course. It is not part of the list, because it cannot be written as decimal number. Proof: All decimal numbers that can be written in this form *are already in the list*. And there is no reason why 1/9 should be missing, if it could be a decimal fraction.
> If it were part of the list , then a_inf would be a well-behaved > natural number , but it isn't .
It is neither well behaved nor a natural number. The sequence 0.111... does not exist other than by a finite name or formula.
> Limits only work properly with real numbers, not natural numbers .
Yes, that is true. But (and please read this very attentively!): Cantor's argument requires the existence of the complete sequence 0.111.... in digits:
You can see this easily here:
0.0 0.1 0.11 0.111 ...
when replacing 0 by 1 has an anti-diagonal, the FIS of which are always in the next line. So the anti-diagonal is not different from all lines, unless it has an infinite sequence of 1's. But, as we just saw, this is impossible.