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

On 18.12.2024 11:16, Mikko wrote:

On 2024-12-17 19:29:52 +0000, WM said:
 

On 17.12.2024 14:08, Mikko wrote:

On 2024-12-16 11:04:17 +0000, WM said:
 

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.

 They are dense because there are other intervals between the point and the
interval.

 The distance between intervals (in some location) is finite but much, much larger than the finite length of the interval. This distance is the distance between intervals which are next to each other. Therefore there is nothing in between.

 There is no next interval and therefore no distance to the next interval
as there are always other intervals nearer.

 Mathematics says the covering by intervals is 3/oo. Therefore the ratio between not covered part and covered part of the positive real axis is oo/3. That implies an average of oo/3 which in some locations must be realized.

That proves the existence of distances between next intervals much larger than the finite lengths of the intervals.

It excludes density of intervals.