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NNTP-Posting-Date: Fri, 15 Nov 2024 18:24:32 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Fri, 15 Nov 2024 10:24:38 -0800
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On 11/15/2024 10:05 AM, Ross Finlayson wrote:
> On 11/15/2024 09:55 AM, Jim Burns wrote:
>> On 11/15/2024 5:10 AM, WM wrote:
>>> On 14.11.2024 19:31, Jim Burns wrote:
>>>> On 11/14/2024 5:20 AM, WM wrote:
>>
>>>>> Therefore
>>>>> a geometric representation let alone proof of
>>>>> most of Cantor's bijections is impossible.
>>>>
>>>> Consider geometry.
>>
>>>> For two triangles △A′B′C′ and △A″B″C″
>>>> if
>>>> △A′B′C′ and △A″B″C″ are similar triangles
>>
>>>> then
>>>> corresponding sides are in the same ratio
>>
>>>>> Therefore
>>>>> a geometric representation let alone proof of
>>>>> most of Cantor's bijections is impossible.
>>>
>>> Your writing is unreadable
>>
>> A geometric representation of
>> square.root, multiplication, and division exist.
>> One representation uses similar triangles.
>>
>> Also, a geometric representation of
>> addition, subtraction, and order exist.
>>
>> Cantor's bijection ⟨i,j⟩ ↦ k ↦ ⟨i,j⟩
>> ⎛ k = (i+j-1)⋅(i+j-2)/2+i
>> ⎜ i = k-⌈(2⋅k+¼)¹ᐟ²-1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-3/2⌉/2
>> ⎝ j = ⌈(2⋅k+¼)¹ᐟ²+1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-1/2⌉/2-1-k
>> is composed of
>> square.root, multiplication, division, addition,
>> subtraction, and ⌈ceiling⌉ (order),
>> for all of which geometric representations exist.
>>
>>> but that does not matter because
>>> of course only a disproof is possible,
>>> since there are no bijections.
>>
>> After all bijections are excluded,
>> of course there are no bijections.
>>
>> On the other hand,
>> ⎛ k = (i+j-1)⋅(i+j-2)/2+i
>> ⎜ i = k-⌈(2⋅k+¼)¹ᐟ²-1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-3/2⌉/2
>> ⎝ j = ⌈(2⋅k+¼)¹ᐟ²+1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-1/2⌉/2-1-k
>> exists.
>>
>>>> Setting aside for a moment
>>>> what you _think_ Cantor's bijection is,
>>>> what part of _that_
>>>> is impossible to represent geometrically?
>>>
>>> It is impossible to cover the matrix
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ...
>>> by shuffling, shifting, reordering the X,
>>> because they are not distinguishable.
>>
>> ⟨k,1⟩ ↦ ⟨i,j⟩ ↤ ⟨k,1⟩
>>
>> ⎛ i = k-⌈(2⋅k+¼)¹ᐟ²-1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-3/2⌉/2
>> ⎜ j = ⌈(2⋅k+¼)¹ᐟ²+1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-1/2⌉/2-1-k
>> ⎝ k = (i+j-1)⋅(i+j-2)/2+i
>>
>> Each ⟨k,1⟩ sends X to ⟨i,j⟩
>> Each ⟨i,j⟩ receives X from ⟨k,1⟩
>>
>> According to geometry.
>> Which I predict makes geometry wrong[WM], too.
>>
>>
>
> Non-standard models of integers exist.
>
>
> Russell's retro-thesis "ordinary infinity"
> is sort of a lie - if there's infinity
> it's extra-ordinary - somebody like Dana Scott
> had introduced "circle" and "box" modalities,
> because he was into modal logic and relevance logic,
> and "each" is not always "all".
>
> Of course it's well-known that any mere stipulation
> is formally refutable, rather trivially. That
> doesn't excuse absence of reason by any means.
>
> "Statistics" does not "predict", though
> "guesses" I suppose may be said -
> retro-troll.
>
>

"A restriction of comprehension is not a truth."