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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sat, 14 Dec 2024 10:12:42 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Fri, 13 Dec 2024 20:19:24 +0100 schrieb WM:
> On 13.12.2024 19:29, joes wrote:
>> Am Fri, 13 Dec 2024 18:20:59 +0100 schrieb WM:
>>> On 13.12.2024 15:41, joes wrote:
>>>   > Am Fri, 13 Dec 2024 09:42:36 +0100 schrieb WM:
>>>
>>>   >> The subset is considered as its own independent set D = {10n | n
>>>   >> ∈ ℕ}
>>>   >> and then it is attached to the set ℕ = {1, 2, 3, ...}. That does
>>>   >> not change the subset.
>>>   > It changes the domain from D to N. What operation is „attachment”?
>>> You can also say pairing. The elements of D are paired with the
>>> elements 10n of ℕ. After this small detour everything proceeds as
>>> usual.
>> Aha, and what is paired with the rest of N? This is not the identity.
>> I was under the impression the bijection step came later.
> No, it came earlier and it was no bijecton.
I mean, you apply your function before the bijection, but then also
change the domain of the bijection to be N instead of D, which of
course doesn’t preserve it, as there is nothing to be paired with
the nondivisible numbers. NB this does NOT mean there aren’t „enough”
multiples of 10 (there’s one for every natural, obviously); they HAVE
already been paired on the other side. Non-multiples of 10 are only
members of ONE of the sets being bijected - N, not D. Do not confuse them.

-- 
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.