On 1 Mai, 14:26, Dan <dan.ms.ch...@gmail.com> wrote: > On May 1, 2:55 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 1 Mai, 13:15, Dan <dan.ms.ch...@gmail.com> wrote: > > > > Do you think the natural numbers *can be enumerated*? > > > No. Look here for a detailled exposition: > > >http://arxiv.org/ftp/math/papers/0305/0305310.pdf > > If I knew it were possible for 4 to follow 2 in the natural numbers , > I would be worried to use infinity . The difference between a 'unknown > infinity' and 'the infinity of natural numbers' , is that the latter > is determined, by thought , and within thought . There is no bluing, > no information missing, no indeterminacy . > 2 is always followed by 3 . > 999 is always followed by 1000 .
Correct. > > If I thought of a formula for an infinite sequence of digits, I can't > pretend "not to know the digits" , or the formula , when communicating > with myself .
There are many such formuas: 1/9, pi. > > I've honestly tried to follow the argument in section 5 , but it's > incomprehensible .
It is easy. Look at the sequence 0.1 0.11 0.111 ... Each term has only natural indices. The limit 0.111... has not only natural indices, because all natural indices are in the terms of the sequence. So no index or set of indices remains to distinguish 0.111... from all terms of the sequence.
Of course you can distinguish every term of the sequence from a larger one. And this is somethimes used as an argument, that 0.111... can be distinguished from every term of the sequence. Erroneously! Since, if the limit could be written by digits at finite places only, it was a term of the list.
The same argument is used in paragraph 5. Every term like 0.111 can be reflected at the decimal point, yielding 111.0, since every power 10^-n has a reciprocal 10^n, *as lomg as n is a natural number*. 0.111... cannot be reflected at the decimal point. This imples that there are not only natural numbers as indices or exponents.
I use only this theorem:
As long as n is a natural number, 10^-n is as well defined as 10^n.