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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sat, 28 Dec 2024 09:15:35 -0500
Organization: i2pn2 (i2pn.org)
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On 12/28/24 8:48 AM, WM wrote:
> On 28.12.2024 06:23, Moebius wrote:
>>
>>>> On 12/17/24 4:51 PM, WM wrote:
>>
>>>>> all FISONs contain/are all n.
> 
>> Hint: For each and every n e IN there is a FISON such that n is in it.
> 
> No. If so, then the union of all FISONs would contain all n. But fact is 
> that the union of all FISONs is a FISON (all FISONs have infinitely many 
> successors) and leaves almost all numbers outside
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> 
> Regards, WM

No, doing an INFIITE union (which a union of all FISONs would be) can 
result in something different in type then the union of a finite number 
of FISONs.

Just like the combining of ALL natural numbers gives us an infinite set, 
while ony combining a finite number of natural numbers gives us a finite 
set.

You logic just can't handle infinite operations, and thus can't actually 
have the set of Natural Numbers as an element of it,