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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sun, 2 Mar 2025 19:42:10 -0500
Organization: i2pn2 (i2pn.org)
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On 3/2/25 1:05 PM, WM wrote:
> On 02.03.2025 13:22, Richard Damon wrote:
>> On 3/1/25 1:18 PM, WM wrote:
>>> On 01.03.2025 17:25, Richard Damon wrote:
>>>> On 3/1/25 6:58 AM, WM wrote:
>>>
>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo. Note for all! The limit 
>>>>> cannot exist without bridging this infinite gap.
>>>>>
>>>> But what is "n" in the above definition?
>>>
>>> It is a natural number that we can represent in principle by making 
>>> as many strokes.
>>
>> SO, Which Natural Number can it not be that makes N_def different than N?
> 
> They cannot be expressed, neither by stroks nor by digits. Their 
> existence can only be proved: UF = ℕ ==> Ø = ℕ.

So, what numbers can't be expressed?

What is the highest expressable number?

If there isn't one, why not?

This is the flaw in your logic, you think there are two different 
classes of the infinite set of Natural Numbers, one "defined" that can't 
have a highest (as what keeps us from defining the next number) and then 
you need to invent something higher to hide the fact that your first set 
is actually an infinite set of all the Natural Numbers, that you Naive 
Logic just can't handle

>>>> Note, the value AT the limit, and the values appraching the limit of 
>>>> things can be different.
>>>
>>> The cannot differ by a fixed quantity like ℵo.
>>
>> Sure a limit can. What says it can't?
> 
> If it differes then it is not a limit by a matheologial credo in absurdum.
> 
> Regards, WM
> 
> 
> 

And you are the one trying to take that limit based on your naive 
mathological operations.

Yes, when we talk about "in the limit of completing the set" it is a 
different sort of operation than the normal mathematics limit, as we get 
infinities that of course never change "value". That is why we don't 
write it as a normal mathematics limit. N is not limit n-> inf of F(n), 
but the set of all Natural Numbers with the appropriate set of properties.