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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Wed, 04 Dec 2024 06:53:29 -0500
Organization: Peripheral Visions
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Moebius presented the following explanation :
> Am 04.12.2024 um 12:26 schrieb Ben Bacarisse:
>> FromTheRafters <FTR@nomail.afraid.org> writes:
>> 
>>> Moebius expressed precisely :
>>>> Am 04.12.2024 um 02:02 schrieb Moebius:
>>>>> Am 04.12.2024 um 01:47 schrieb Chris M. Thomasson:
>>>>>> On 12/3/2024 2:32 PM, Moebius wrote:
>>>>>>> Am 03.12.2024 um 23:16 schrieb Moebius:
>>>>>>>> Am 03.12.2024 um 22:59 schrieb Chris M. Thomasson:
>>>>>>>
>>>>>>>>> However, there is no largest natural number, when I think of that I
>>>>>>>>> see no limit to the naturals.
>>>>>>>
>>>>>>> Right. No "coventional" limit. Actually,
>>>>>>>
>>>>>>>  ����� "lim_(n->oo) n"
>>>>>>>
>>>>>>> does not exist.
>>>>>>
>>>>>> In the sense of as n tends to infinity there is no limit that can be
>>>>>> reached [...]?
>>>>> Exactly.
>>>>> We say, n is "growing beyond all bounds". :-P
>>>>
>>>> On the other hand, if we focus on the fact that the natural numbers are
>>>> sets _in the context of set theory_, namely
>>>>
>>>>         0 = {}, 1 = {{}}, 2 = {{}, {{}}}, ...
>>>>
>>>> =>      0 = {}, 1 = {0}, 2 = {0, 1}, ...
>>>>
>>>> (due to von Neumann)
>>>>
>>>> then we may conisider the "set-theoretic limit" of the sequence
>>>>
>>>>        (0, 1, 2, ...) = ({}, {0}, {0, 1}, ...).
>>>>
>>>> This way we get:
>>>>
>>>>        LIM_(n->oo) n = {0, 1, 2, ...} = IN. :-P
>>>>
>>>> I'd like to mention that "lim_(n->oo) n" is "old math" (oldies but
>>>> goldies) while "LIM_(n->oo) n" is "new math" (only possible after the
>>>> invention of set theory (->Cantor) and later developments (->axiomatic
>>>> set theory, natural numbers due to von Neumann, etc.).
>>>
>>> If you say so, but I haven't seen this written anywhere.
>
> @FromTheAfter: https://en.wikipedia.org/wiki/Set-theoretic_limit

I still don't see it. Don't get me wrong here, I like the capitalized 
LIM for set limit, it helps clarity in case one forgets the context is 
set theory.