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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sat, 18 Jan 2025 09:41:44 +0100
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On 18.01.2025 00:08, Jim Burns wrote:
> On 1/17/2025 2:40 PM, WM wrote:
>> Am 17.01.2025 um 17:53 schrieb Jim Burns:
>>> On 1/17/2025 4:08 AM, WM wrote:
>>>> On 16.01.2025 23:22, Jim Burns wrote:
> 
>>>>> Nowhere,
>>>>> among what appears and
>>>>> among what doesn't appear,
>>>>> is there finite ω-1 and infinite (ω-1)+1
>>>>
>>>> So it appears because ω and ω-1 are dark.
>>>
>>> We never see ω and ω-1
>>> We see descriptions of ω and ω-1
>>> That is sufficient for knowledge of ω and ω-1
>>
>> Dark numbers cannot be seen,
>> if you understand by that phrase
>> be put in a FISON.
> 
> Definitions can be seen.

Yes, dark numbers however can be handled only collectively. That 
distinguishes them from visible numbers.
> 
> Finite sequences of claims, each claim of which
> is true.or.not.first.false
> can be seen.

Like the visible numbers.
> 
> ----
> The finite extends
> much further than you (WM) think it does.
> Infinitely further than you think it does.

No. As long as you deny Bob's existence and violate logic you are not a 
reliable source.

> No finite ordinal has
> an infinite immediate successor.

Maybe. But then there is no infinite ordinal.

>>> You (WM) introduce
>>> negative cardinality (darkᵂᴹ numbers)
>>> in an attempt to fit these claims together.
>>
>> No, I don't.
> 
> I'm willing to believe that
> you didn't intend to introduce negative cardinality.
> Nonetheless, you did.
> 
> A potentiallyᵂᴹ infinite set is larger.than.any.finite.

No, it is a finite step in a process with no upper bound.
> 
> An actuallyᵂᴹ infinite set A isn't potentially infinite.
> It isn't larger.than.any.finite.

It is much larger. Every finite is set is infinitesimal compared to ω, 
namely it is smaller than ω/n for every visible number n.

> There is a larger finite set F.

No.
> 
> Actuallyᵂᴹ infinite A has
> a potentiallyᵂᴹ infinite subset P

Yes, but better this is expressed as subcollection or a multitude of 
finite sets.

Regards, WM