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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (conserved infinity)
Date: Wed, 20 Nov 2024 17:00:05 -0800
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On 11/20/2024 4:02 PM, Ross Finlayson wrote:
> On 11/20/2024 12:23 PM, FromTheRafters wrote:
>> on 11/20/2024, Ross Finlayson supposed :
>>> On 11/20/2024 12:05 PM, FromTheRafters wrote:
>>>> WM wrote on 11/20/2024 :
>>>>> On 20.11.2024 15:14, FromTheRafters wrote:
>>>>>> WM formulated on Wednesday :
>>>>>
>>>>>>>
>>>>>>> It does not make it wrong, but it unmasks it at imprecise. That's
>>>>>>> why I don't like it. We can do better.
>>>>>>
>>>>>> It works well enough.
>>>>>
>>>>> Really? Then you can answer the following questions:
>>>>>
>>>>> Let every unit interval after a natural number on the real axis be
>>>>> coloured white with exception of the powers of 2 which are coloured
>>>>> black. Is it possible to shift the black intervals so that the whole
>>>>> real axis becomes black?
>>>>
>>>> No, of course not.
>>>>
>>>>> Or: Let every unit interval after a natural number on the real axis be
>>>>> coloured as above with exception of the intervals after the odd prime
>>>>> numbers which are coloured red. Is it possible to shift the red
>>>>> intervals so that the whole real axis becomes red?
>>>>
>>>> No, of course not.
>>>>
>>>>> What colour has the real axis after you have solved both tasks?
>>>>
>>>> Depending on the order of the tasks. I think half red or half black.
>>>
>>> Well you have to reference academic reference and describe "supertask"
>>> besides "asymptotics" about where "the asymptotic density of black or
>>> red respectively is 1 in the limit", that you point to "supertask"
>>> instead of mumbling like it's not already considered by proper minds,
>>> not just ketonic neck-flap gaspers of having failures altogether
>>> of any sort of related-rates problems.
>>>
>>> This is mathematics: humor is irrelevant, and so is what
>>> anybody "thinks", or, "feels".
>>>
>>> It only matters what "is", and there's a language of it,
>>> so use it. (Or lose it.)
>>>
>>> Good sir
>>
>> If painted black and then red, it will be red. If painted red and then
>> black, it will be black. These are real intervals, and as such I assume
>> real powers of two. In both scenarios, none of the negative real axis is
>> at all affected.
> 
> "Restricted Sequence Element Interchange" is an idea that
> is a sort of "conservation principle" about things in an
> Integer Continuum or Linear Continuum, here for example
> an Integer Continuum. The idea is that any switch, as much
> as it changes a plain 0101 to 0011, happens once-at-a-time
> or the pair-wise, about basically, "after so much time given
> to find an offset to exchange and another for its place,
> and to update the state of the data structure that it is so,
> that it's a matter of book-keeping and related-rates or
> a system of algorithmic resources in numerical resources,
> and time", that it's not merely giving x_infinity when
> "at time 0 < Sum 1/n^2 < 1 that element n changes from
> 0 to 1" that at t_oo at n = oo that it's all 1's,
> that it's so asymptotically, or that the density as
> always filling in closer to the origin has that any
> first different is arbitrarily far away, still has
> that it's an honest account of book-keeping to make
> that into a structure as if you had to implement it
> and more than merely a lazy, forgetful mathematician's
> exercise in induction that can easily arrive at
> from 010101... to 00000... or 111111....
> 
> 
> Anyways there's a theory about these things that
> basically make for cases besides those that just
> shove off the end and put it off forever, besides
> the "asymptotics" is what's called "supertasks".
> 
> These may include for systems that are merely
> "very, very large" when not "actually infinite",
> that some practical or effective infinity, yet
> results as a "point at infinity" which is a critical
> or accumulation point, for the swapped-out items.
> 
> Like a "point at infinity", a "prime at infinity". Or not,
> it's among things entirely independent standard number
> theory, which some have as that the integers don't actually
> have a standard model anyways, only fragments and extensions.
> 
> Anyways these sorts of things make for reasonings when
> things exchange and conserve besides one-sidedly shove off.
> 
> 

A prime at infinity? Keep in mind that there is an infinite number of 
primes. So, are you talking about perspective as in a point at infinity?