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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Wed, 19 Mar 2025 20:19:21 +0100
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On 19.03.2025 16:18, Jim Burns wrote:
> On 3/18/2025 12:58 PM, WM wrote:

> ⎛ Bob is in.matrix before.swaps, but
> ⎝ Bob is NOT in.matrix after.swaps.

Then he must have left by a swap or by a miracle of matheology. I do not 
accept the latter, ad I can prove that the former is in contradiction to 
logic.>
>>> After all the swaps,
>>> without ever disappearing into
>>> anywhere other than a finitec set,
>>> Bob disappears out of all finite sets.
>>
>> No, that is impossible.
> 
> Then there are subsets larger than their sets.

No, there are dark numbers. That is enough.
> 
>> If there is an "after all swaps",
>> then all O have settled within the matrix.
> 
> That matrix,
> with rows and columns indexed by
> the emptiest inductive set,
> after all the swaps,
> that matrix doesn't hold any O.

That is wrong because O cannot get lost. The O remain but at dark places.
> 
> ⎛ Assume otherwise.
> ⎜ Assume that it's after all swaps,
> ⎜ and O is in cell <k,1>

That's the first mistake. All O's are in dark cells which cannot be 
distinguished.
>>> After all the swaps,
>>> Bob is lost from all finite indices.
>>
>> No, that is impossible.
>> Here lies your mistake.
>> The matrix has no drain!
> 
> If the set of
> all set.sizes needing a drain
> is a set of a size needing a drain,
> then
> that set has subsets larger than it, the set.
> 
That is impossible. Therefore dark numbers must be accepted.
Regards, WM