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

On 2024-12-14 08:53:19 +0000, WM said:

On 14.12.2024 09:41, Mikko wrote:

On 2024-11-19 11:04:08 +0000, WM said:
 

On 19.11.2024 10:32, Mikko wrote:

On 2024-11-18 14:29:40 +0000, WM said:
 

On 18.11.2024 10:58, Mikko wrote:

On 2024-11-17 12:46:29 +0000, WM said:

 

There are 100 intervals for each natural number.
This can be proven by bijecting J'(100n) and J(n). My intervals are then exhausted, yours are not.

 Irrelevant.

 Very relevant.

 It is not relevant if no relevancy is shown.

 But if relevancy is only deleted, it can show up again:
 Every finite translation of any finite subset of intervals J(n) maintains the relative covering 1/5. If the infinite set has the relative covering 1 (or more), then you claim that the sequence 1/5, 1/5, 1/5, ... has limit 1 (or more).

 There is a bijection between your J and my J', where
J'(n) = (n/100 - 1/10, n/100 + 1/10): for each n there
is one interval J(n) and one interval of J'(n). Whateever
you infer from that is either an invalid inference or
a true conclusion.

Please refer to the simplest example I gave you on 2024-11-27:
The possibility of a bijection between the sets ℕ = {1, 2, 3, ...} and D = {10n | n ∈ ℕ} is contradicted because for every interval (0, n] the relative covering is not more than 1/10, and there are no further numbers 10n beyond all natural numbers n.

It is already proven that there is such bijection. What is proven cannot
be contradicted undless you can prove that 1 = 2.

The sequence 1/10, 1/10, 1/10, ... has limit 1/10.

Irrelevant as the proof of the exitence of the bijection does not
mention that sequence.
--
Mikko