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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Mon, 3 Mar 2025 10:10:51 +0100
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On 03.03.2025 01:42, Richard Damon wrote:
> On 3/2/25 1:05 PM, WM wrote:

>>> 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?

Dark numbers.
> 
> What is the highest expressable number?

That does not exist because with n also n+1 is expressable.
> 
> If there isn't one, why not?

We call that phenomenon potential infinity.
> 
> 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,

It can be proved. All infiite sets as produced by Peano, Zermelo, or v. 
Neumann are potentially infinite, namely produced by inductive reasoning 
with subsequent quantifier exchange. In exactly the same way as Z₀ is 
constructed by its elements, the set of removable FISONs is constructed 
by its elements. The result is Z₀ = ℕ  ==> Ø = ℕ.

>>>>> Note, the value AT the limit, and the values appraching the limit 
>>>>> of things can be different.
>>>>
>>>> They cannot differ by a fixed quantity like ℵo.
>>>
>>> Sure a limit can. What says it can't?
>>
There are dark numbers collected.
> Yes, when we talk about "in the limit of completing the set" it is a 
> different sort of operation than the normal mathematics limit,

It adds the dark numbers to the inductive set.

> 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), 

Right. You try to collect the dark numbers without mentioning it.

Regards, WM