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

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 dpend on how you translate.
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. Your "impossible" is false.
--
Mikko