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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 19:05:54 -0800
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On 11/20/2024 5:18 PM, Ross Finlayson wrote:
> On 11/20/2024 05:00 PM, Chris M. Thomasson wrote:
>> 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?
> 
> What about it?
> 
> Number theorists have various ways to define a point
> at infinity, vis-a-vis geometry's usual notion as
> of a perspective point-at-infinity, and projective
> geometry's projective point-at-infinity, and number
> theorist's compactification of the naturals or variously
> with regards to the fundamental theorem of arithmetic,
> whether it's so at infinity, or not.
> 
> It is rather regarded that these notions are _significant_
> and _relevant_ and as well that they're _independent_,
> usual enough fragments of theories of fragments of models
> of numbers, or fixed views, and these kinds of things.
> 
> So, there are models of integers with a "prime at infinity",
> i.e. its only multiplicative factors are itself and 1,
> and it's defined. There are others where it's composite,
> for example being a product of each of the primes, there
> are others, there are each the others. A "prime" at infinity
> decides some things and makes an _opinion_, it's a _singular_
> of what's a _multiplicity_ of models.
> 
> It's usually said to be "number theory's" and no other
> theories of mathematics thusly get any say at all about it,
> unless said theory includes all the multiplicity of its
> considerations. In this sense a usual nominalist fictionalist
> "we can't say nothing" may be a good thing, as they can't say
> anything wrong about it either, yet "you can't say nothing, either"
> is considered objectionable, because number theorists can and do.
> 
> Consider double primes: two primes, separated by two, for
> example, 11 and 13, or, 17 and 19. Now, there are many, many
> less, of the double primes, than there are of primes. Then,
> for triple primes, there's only 2, 3, 5. There are no other
> (known) finite triple primes. Yet at infinity, it could
> be in or among double, triple, even quadruple primes,
> even n-tuple primes, in various models of integers.
> 
> So, anyways, the projective and perspective are very
> relevant geometry beyond renderings, and about it.
> 
> 

A point at infinity tends to be finite for the perspective?