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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 05:29:28 -0500
Organization: Peripheral Visions
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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 = {{}, {{}}, ...

Typo, needs another closing curly bracket.

> =>    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.