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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Tue, 17 Dec 2024 07:34:14 -0500
Organization: i2pn2 (i2pn.org)
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On 12/16/24 6:28 AM, WM wrote:
> On 16.12.2024 11:23, Mikko wrote:
>> On 2024-12-15 11:33:15 +0000, WM said:
> 
>>>
>>> We cannot name dark numbers as individuals.
>>
>> We needn't. The axioms of natural numbers ensure that every natural 
>> number
>> has a successor,
> 
> The set, i.e. all numbers together, has no successor. That is a 
> necessary condition for using all with no exception. That must happen 
> according to
> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}.
> 
>> If that is not possible then there are no
>> natural numbers.
> 
> That is not possible for an actually infinite set. It is only possible 
> for numbers coming into being.
> 
> Regards, WM
> 
> 

No, if you actually have that actually infinite set, they are all there 
to begin with, but of course, you can't make that infinite set with your 
finite logic, so you just can't handle such a set.