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

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.

That's what "dense" means.

Yes that is the meaning of "dense". There is no finite distance between next points, e.g. rationals. Therefore the intervals are not dense. Therefore the intervals do not cover all rational points.
Regards, WM