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

On 24.11.2024 13:56, Mikko wrote:

On 2024-11-22 11:05:58 +0000, WM said:
 

On 22.11.2024 09:52, Mikko wrote:

On 2024-11-21 19:22:39 +0000, WM said:

>

True but irrelevant because it may be even closer to the end of
another interval.

>
Every end of any interval is irrational.

>
Irrelevant to the claim that a point outside of all intervals has two
nearest intervals

>
What else should it have?

 For every point that is neither an endpoint nor interior point of any
of the intervals there is no nearest interval because between that point
and any interval there is another interval.

Wrong. Starting from that point a cursor, by simplest logic, will either touch nothing or something. If it touches something, it can only be an irrational endpoint.

That is concerned by the fact that in the complement of measure oo - 3 rational numbers must exist.

 As every rational number is the midpoint of one of the intervals it is
obvious that no rational is outside of all intervals.

Just this is contradicted by the infinite measure. Cabtor cannot count all rationals.

Let me know what else than an irrational number could be next to a point in the complement, if there were no rational numbers.

 No point is next to any other point because there are other points between
the two.

In fact. But that would be impossible if all rationals were hidden behind irrational endpoints.
Regards, WM