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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 12:42:23 +0100
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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

> 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!

Same with the notions of /bijections/. Explained in his book but denied 
by WM these days.

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