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

On 16.12.2024 10:51, Mikko wrote:

On 2024-12-16 08:55:39 +0000, WM said:
 

On 15.12.2024 22:14, Richard Damon wrote:

On 12/15/24 2:29 PM, WM wrote:

>

Next is a geometric property, in particular since the average distance of intervals is infinitely larger than their sizes.

>

Not sure where you get that the "average" distance of intervals is infinitely larger than ther sizes.

>
The accumulated size of all intervals is less than 3 over the infinite length.

 True.
 

Hence

 False.
 

there is at least one location with a ratio oo between distance to the interval and length of the interval.

 False. Regardless which interval is "the" interval the distance to that
interval is finite and the length of the interval is non-zero so the
ratio is finite.

Well, it is finite but huge. Much larger than the interval and therefore the finite intervals are not dense.>

 Start there with the cursor. It will hit one next interval. Crash.

 No, it does not. It does not touch an interval before passing another
interval.

That is nonsense, because the distance, at least at one location, is much larger than the finite interval. That proves that, at that location, the intervals are not dense. That proves that not all rationals are included in intervals.

An interval it touches after passing other intervals is not
the next interval.

The multiple of a finite length (of an interval) does not suffer from intervals showing up from behind.
Regards, WM