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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Wed, 2 Apr 2025 19:45:39 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Wed, 02 Apr 2025 16:07:20 +0200 schrieb WM:
> On 27.03.2025 22:34, Alan Mackenzie wrote:
>> WM <invalid@no.org> wrote:
>>> Am 26.03.2025 um 22:34 schrieb Alan Mackenzie:
>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>>>> On 26.03.2025 07:24, Jim Burns wrote:

>>>>> "WM-logic" says that lossless exchanges at finite steps are lossless
>>>>> ....
>>>> Nobody has contradicted this, that I'm aware of.  It's the
>>>> accumulation of _all_ of these lossless exchanges where unexpected
>>>> things happen.
>>> If they happen, then there is a first instance where they happen.
>>> Every n.e. subset of a countable set has a first element.
>> The set of integer steps at which a loss occurs is empty.
> There are no other steps at which anything could occur.
There is the limit.

>>  It thus has no least member.
> Nevertheless all members are finite integers, and afterwards nothing
> happens anymore.
After what exactly?

>> It is only in the infinite limit where the loss occurs.
> Bijections have no limit.
Good that you're finally coming around on this.

> "The infinite sequence thus defined has the peculiar property to contain
> the positive rational numbers completely, and each of them only once at
> a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
> Limits are not determined places.
Why should you think "there" is a rational number?

>> In the limit, it passes _all_ places.
>>   In informal language, it "disappears off to infinity",
> There is no chance to disappear. And never infinity is reached.
"Disappear" is just a figure of speech, like running out of view.
Infinity is "reached" in - wait for it - the limit.

>> and thus is no longer at one of the numbered places.
>> You base your mathematical thinking on faulty intuition.  You do not
>> base it on the axioms and logic which have chrystallised out of a lot
>> of very clever thinking over the last few centuries.
> Do you think that Cantor was wrong?
Aren't you the one?

-- 
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.