Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 2024-11-14 10:34:52 +0000, WM said:

On 14.11.2024 10:17, Mikko wrote:

On 2024-11-13 16:14:02 +0000, WM said:
 

On 13.11.2024 11:39, Mikko wrote:

On 2024-11-12 13:59:24 +0000, WM said:

 

Cantor said that all rationals are within the sequence and hence within all intervals. I prove that rationals are in the complement.

 He said that about his sequence and his intervals. Infinitely many of them
are in intervals that do not overlap with any of your J(n).

 The intervals J(n) = [n - 1/10, n + 1/10] cover the relative measure 1/5 of ℝ+. By translating them to match Cantor's intervals they cover ℝ+ infinitely often. This is impossible. Therefore set theorists must discard geometry.

 The intervals J(n) are what they are. Translated intervals are not the same
intervals. The properties of the translated set depend on how you translate.

 No. Covering by intervals is completely independent of their individuality and therefore of their order.

Translated intervals are not the same as the original ones. Not only their
order but also their positions can be different as demonstrated by your
example and mine, too.

Therefore you can either believe in set theory or in geometry. Both contradict each other.

Geometry cannot contradict set theory because there is nothing both
could say. But this discussion is about set theory so geometry is not
relevant.

For example, if you translate them to J'(n) = (n/100 - 1/10, n/100 + 1/10)
then the translated intervals J'(n) wholly cover the postive side of the
real line.

 By shuffling the same set of intervals which do not cover ℝ+ in the initial configuration, it is impossible to cover more. That's geometry.

So what part of ℝ+ is not covered by my J'?
--
Mikko