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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 11:26:06 +0000
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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 = {{}, {{}}, ...
>
> 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.

It's usually framed in terms of least upper bounds, so that might be why
you are not recalling it.

Ironically, there is a very common example of a "set theoretic limit"
which is the point-wise limit of a sequence of functions.  Since
functions are just sets of pairs, these long-known limits are just the
limits of sequences of sets.  It's ironic because WM categorically
denies that /any/ non-constant sequence of sets has a limit, yet the
basic mathematics textbook he wrote includes the definition of the
point-wise limit, as well as stating that functions are just sets of
pairs.  He includes examples of something he categorically denies!

-- 
Ben.