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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Tue, 18 Mar 2025 17:58:09 +0100
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On 18.03.2025 17:34, Jim Burns wrote:
> On 3/18/2025 5:00 AM, WM wrote:

>> Bob can only disappear in a set of finite size
>> because all exchanges appear at finite sizes.
> 
> Bob disappears from (is swapped from)
> each finite set A.
> Bob disappears to (is swapped to)
> a different finite set

Yes. But he never leaves the matrix!
> After all the swaps,
> without ever disappearing into
> anywhere other than a finite set,
> Bob disappears out of all finite sets.

No, that is impossible. If there is an "after all swaps", then all O 
have settled within the matrix. Lossless exchanges with losses are not 
allowed.
> 
> The swaps are ⟨n⇄n+1⟩ for all n,
> in order by n,
> in the emptiest inductive set.

Yes, for all n reachable by induction.
> 
>>> The set of all finite.sizes is same.sized as
>>> the set of all finite sizes and Bob.
>>
>> Give the index where Bob is lost.
> 
> Before all the swaps,
> Bob is not lost from all finite indices.

And all his infinitely many colleagues are also not lost from all finite 
indices.
> 
> After all the swaps,
> Bob is lost from all finite indices.

No, that is impossible. Here lies your mistake. The matrix has no drain! 
All X destinated to be exchanged with O are placed inside the matrix at 
the beginning.
> 
> There is no finite index,
> either visible or dark,
> from which Bob is last.lost,
> or to which Bob is last.lost.

That means he cannot get lost. All motions happen at finite indices. He 
remains in the matrix, but only the X's are visible after all:
XOOO... XXOO... XXOO... XXXO... ... XXXX...
XOOO... OOOO... XOOO... XOOO... ... XXXX...
XOOO... XOOO... OOOO... OOOO... ... XXXX...
XOOO... XOOO... XOOO... OOOO... ... XXXX...
...............................................................................

Regards, WM

Regards, WM