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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Tue, 3 Dec 2024 13:59:08 -0800
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On 12/3/2024 3:35 AM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
> 
>> On 12/2/2024 4:00 PM, Chris M. Thomasson wrote:
>>> On 12/2/2024 3:59 PM, Moebius wrote:
>>>> Am 03.12.2024 um 00:58 schrieb Chris M. Thomasson:
>>>>> On 12/2/2024 3:56 PM, Moebius wrote:
>>>>>> Am 03.12.2024 um 00:51 schrieb Chris M. Thomasson:
>>>>>>> On 12/1/2024 9:50 PM, Moebius wrote:
>>>>>>>> Am 02.12.2024 um 00:11 schrieb Chris M. Thomasson:
>>>>>>>>> On 11/30/2024 3:12 AM, WM wrote:
>>>>>>>>
>>>>>>>>>> Finite initial segment[s]: F(n) = {1, 2, 3, ..., n}    (n e IN).
>>>>>>> [...]
>>>>>>>
>>>>>>> When WM writes:
>>>>>>>
>>>>>>> {1, 2, 3, ..., n}
>>>>>>>
>>>>>>> I think he might mean that n is somehow a largest natural number?
>>>>>>
>>>>>> Nope, he just means some n e IN.
>>>>>
>>>>> So if n = 5, the FISON is:
>>>>>
>>>>> { 1, 2, 3, 4, 5 }
>>>>>
>>>>> n = 3
>>>>>
>>>>> { 1, 2, 3 }
>>>>>
>>>>> Right?
>>>>
>>>> Right.
>>> Thank you Moebius. :^)
>>
>> So, i n = all_of_the_naturals, then
> 
> You are in danger of falling into one of WM's traps here.  Above, you
> had n = 3 and n = 5.  3 and 5 are naturals.  Switching to n =
> all_of_the_naturals is something else.  It's not wrong because there are
> models of the naturals in which they are all sets, but it's open to
> confusing interpretations and being unclear about definition is the key
> to WM's endless posts.
> 
>> { 1, 2, 3, ... }
>>
>> Aka, there is no largest natural number and they are not limited. Aka, no
>> limit?
> 
> The sequence of FISONs has a limit.  Indeed that's one way to define N
> as the least upper bound of the sequence
> 
>   {1}, {1, 2}, {1, 2, 3}, ...
> 
> although the all terms involved need to be carefully defined.
> 
>> Right?
> 
> The numerical sequence 1, 2, 3, ... has no conventional numerical limit,
> but, again, if the symbols 1, 2, 3 etc stand for sets (as in, say, Von
> Neumann's model for the naturals) then the set sequence
> 
>   1, 2, 3, ...
> 
> does have a set-theoretical limit: N.
> 

However, there is no largest natural number, when I think of that I see 
no limit to the naturals. I must be missing something here? ;^o