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

On 12/16/24 3:41 AM, WM wrote:

On 15.12.2024 21:32, joes wrote:

Am Sun, 15 Dec 2024 12:25:26 +0100 schrieb WM:

On 15.12.2024 11:56, Mikko wrote:

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

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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.

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

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What is proven under false (self-contradictory) premises can be shown to
be false. Here we have a limit of 1/10 from analysis and a limit of 0
from set theory. That shows that if set theory is right, we have 1/10 =
0 ==> 1 = 0 ==> 2 = 1.

Which sequence do you get a limit of 0 from?

 Sorry, the limit of not indexed numbers is 9/10 according to analysis and 0 according to set theory, resulting in 9/10 = 0.

Which shows that one of them is likely wrong.
SInce your method of analysis will also BY ITSELF say that 0 == 1, we see that your method of analysis is what has the problem.

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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.

But the disproof of the bijection does. There is no reason to forbid
that sequence.

That sequence does not appear in the bijection.

 Therefore people were unaware of its failure.
 

No, it shows you don't understand what a bijection is, and didn't follow it.

Regards, WM

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