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From: Ben Bacarisse <ben@bsb.me.uk>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Wed, 04 Dec 2024 10:56:05 +0000
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"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> 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,

Yes, there is no largest natural.  Let's not loose sight of that.

> when I think of that I see no
> limit to the naturals. I must be missing something here? ;^o

It's just that there are lots of kinds of limit, and a limit is not
always in the set in question.  Very often, limits take us outside of
the set in question.  R (the reals) can be defined as the "smallest" set
closed under the taking of certain limits -- the limits of Cauchy
sequences, the elements of which are simply rationals.

Even if we don't consider FISONs, we can define a limit (technically a
least upper bound) for the sequence 1, 2, 3, ...  It won't be a natural
number.  We will have to expand our ideas of "number" and "size" to get
the smallest "thing", larger than all naturals.  This is how the study
of infinite ordinals starts.

-- 
Ben.