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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Sun, 16 Mar 2025 19:40:43 +0100
Organization: A noiseless patient Spider
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On 16.03.2025 18:19, Jim Burns wrote:
> On 3/16/2025 12:17 PM, WM wrote:
>> On 16.03.2025 13:17, Alan Mackenzie wrote:
>>> WM <wolfgang.mueckenheim@tha.de> wrote:
> 
>>>> ℵo numbers have no FISONs.
>>>
>>> You haven't said what you mean by F.
>>
>> I did in the discussion with JB:
>> F is the set of FISONs.
> 
> And a FISON is
> a Finite Initial Segment Of Naturals.
> However,
> if you (WM) have _non.circularly_ said
> what 'finite' means, I have missed it.

Finite means a natural number. If you don't know try to learn what they 
are. 1 and if n then n+1.
> 
>>> All
>>> natural numbers "have" a FISON
>>
>> Then all natural numbers would be in FISONs.
>> But because of
>> ∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo
>> all FISONs fail to contain all natural numbers.
> 
> You (WM) use an unreliable quantifier swap
> to arrive at that conclusion.

No it is very reliable to say that when all FISONs have infinite 
distance from ω, then no FISON comes closer. Only real cranks can claim 
that this is wrong. But that is not the question. The question is this:

All infinitely many lossless exchanges have finite indices as you 
understand them. Which index, as you understand it, has the first 
lossless exchange where a loss happens (by the way one of infinitely 
many which are required to empty the matrix of O's)?

Regards, WM