Re: Incompleteness of Cantor's enumeration of the rational numbers

On 27.11.2024 10:52, Mikko wrote:

On 2024-11-26 11:05:30 +0000, WM said:
 

On 26.11.2024 10:08, Mikko wrote:

On 2024-11-25 13:55:57 +0000, WM said:

>

But before touching a rational it will touch an irrational.

>
Of course as the starting point is outside of all the intervals and
every rational is in some of the intervals and therefore must be
irrational. But when it has moved to another point it has already
moved over both infinitely many irrationals

>
This is true in every case. The intermediate numbers cannot be discerned. They are dark. This is so in fact between every pair of discernible real numbers: There are infinitely many dark numbers between them.

 Some of the intermediate numbers can be expressed with a finite string.

But most cannot.

In particular, every rational number can.

No. For every unit fraction there exist infinitely many smaller unit fractions, infinitely many of which cannot be expressed. They remain simply to be smaller.
Regards, WM