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

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:
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On 18.11.2024 10:58, Mikko wrote:

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

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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.

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Irrelevant.

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Very relevant.

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It is not relevant if no relevancy is shown.

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But if relevancy is only deleted, it can show up again:
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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. The sequence 1/10, 1/10, 1/10, ... has limit 1/10.
Regards, WM