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

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

On 25.11.2024 13:41, Mikko wrote:

On 2024-11-24 14:19:06 +0000, WM said:

 

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.

 Your "wrong" is wrong. It will either touch something or nothing. If it
touches nothing it cannot touch an irrational endpoint.

 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 and infinitely many
rationals and of course over infinitely many of the intervals.
(Here "the intervals" means the intervals specified by OP.)
--
Mikko