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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Wed, 4 Dec 2024 08:47:45 +0100
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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.).