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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sun, 15 Dec 2024 12:03:27 +0100
Organization: A noiseless patient Spider
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On 14.12.2024 23:45, Richard Damon wrote:
> On 12/14/24 4:34 PM, WM wrote:
>> On 14.12.2024 19:53, Richard Damon wrote:
>>> On 12/14/24 10:55 AM, WM wrote:
>>>> On 14.12.2024 15:08, Richard Damon wrote:
>>>>> On 12/14/24 3:53 AM, WM wrote:
>>>>
>>>>>> Please refer to the simplest example I gave you on 2024-11-27:
>>>>>> The possibility of a bijection between the sets ℕ = {1, 2, 3, ...} 
>>>>>> and D = {10n | n ∈ ℕ} is contradicted because for every interval 
>>>>>> (0, n] the relative covering is not more than 1/10, and there are 
>>>>>> no further numbers 10n beyond all natural numbers n. The sequence 
>>>>>> 1/10, 1/10, 1/10, ... has limit 1/10.
>>>> +
>>>>> Except that we aren't dealng with the FINITE sets of {1, 2, 3, ..., 
>>>>> n} but for the full set of { 1, 2, 3, ... }
>>>>
>>>> Of course. It consist of all finite n. It is the union of all 
>>>> intervals (0, n]. Not more!
>>
>>> But the set of Natural Numbers isn't built that way.
>>
>> You are misinformed. The set of natural numbers is the union of all 
>> FISONs F(n) = {1, 2, 3, ..., n} = (0, n].
>>
>  From where did this concept come from?

The set of FISONs is the set of natural numbers designed by v. Neumann.

https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

Regards, WM