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

Am Mon, 16 Dec 2024 09:41:31 +0100 schrieb WM:

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:

>

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 unless you can prove that 1 = 2.

>
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.

What sequence are you talking about? You can’t compare relative and
absolute measures.

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.

But how does it relate to it?

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.