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NNTP-Posting-Date: Wed, 13 Nov 2024 02:45:51 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (doubling-spaces)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 12 Nov 2024 18:45:56 -0800
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On 11/12/2024 06:22 PM, Ross Finlayson wrote:
> On 11/12/2024 05:38 PM, Chris M. Thomasson wrote:
>> On 11/12/2024 5:24 PM, Ross Finlayson wrote:
>>> On 11/12/2024 05:02 PM, Chris M. Thomasson wrote:
>>>> On 11/12/2024 3:13 PM, Ross Finlayson wrote:
>>>>> On 11/12/2024 01:36 PM, Jim Burns wrote:
>>>>>> On 11/12/2024 12:40 PM, Ross Finlayson wrote:
>>>>>>> On 11/11/2024 12:59 PM, Ross Finlayson wrote:
>>>>>>>> On 11/11/2024 12:09 PM, Jim Burns wrote:
>>>>>>>>> On 11/11/2024 2:04 PM, Ross Finlayson wrote:
>>>>>>>>>> On 11/11/2024 11:00 AM, Ross Finlayson wrote:
>>>>>>>>>>> On 11/11/2024 10:38 AM, Jim Burns wrote:
>>>>>>
>>>>>>>>>>>> Our sets do not change.
>>>>>>>>>>>> Everybody who believes that
>>>>>>>>>>>>   intervals could grow in length or number
>>>>>>>>>>>> is deeply mistaken about
>>>>>>>>>>>>   what our whole project is.
>>>>>>>>>>>
>>>>>>>>>>> How about Banach-Tarski equi-decomposability?
>>>>>>>>>
>>>>>>>>> The parts do not change.
>>>>>>
>>>>>>>> any manner of partitioning said ball or its decomposition,
>>>>>>>> would result in whatever re-composition,
>>>>>>>> a volume, the same.
>>>>>>
>>>>>>> So, do you reject the existence of these?
>>>>>>
>>>>>> No.
>>>>>>
>>>>>> What I mean by "The parts do not change" might be
>>>>>> too.obvious for you to think useful.to.state.
>>>>>> Keep in mind with whom I am primarily in discussion.
>>>>>> I am of the strong opinion that
>>>>>> "too obvious" is not possible, here.
>>>>>>
>>>>>> Finitely.many pieces of the ball.before are
>>>>>>   associated.by.rigid.rotations.and.translations to
>>>>>> finitely.many pieces of two same.volumed balls.after.
>>>>>>
>>>>>> They are associated pieces.
>>>>>> They are not the same pieces.
>>>>>>
>>>>>> Galileo found it paradoxical that
>>>>>> each natural number can be associated with
>>>>>> its square, which is also a natural number.
>>>>>> But 137 is associated with 137²
>>>>>> 137 isn't 137²
>>>>>>
>>>>>> I don't mean anything more than that.
>>>>>> I hope you agree.
>>>>>>
>>>>>>> Mathematics doesn't, ....
>>>>>>
>>>>>> Mathematics thinks 137 ≠ 137²
>>>>>>
>>>>>>
>>>>>
>>>>> 1 = 1^2
>>>>> 0 = 0^2
>>>> [...]
>>>>
>>>> Don't forget the i... ;^)
>>>>
>>>> sqrt(-1) = i
>>>> i^2 = -1
>>>>
>>>> ?
>>>
>>>
>>> Nah, then the quotients according to the
>>> definition of division don't have unique quotients.
>>
>> Do you know that any complex number has n-ary roots?
>>
>>
>>
>>
>>
>> [...]
>
> Consider for example holomorphic functions,
> where there's complex division, thusly,
> it could be a variety.
>
> https://en.wikipedia.org/wiki/Holomorphic_function#Definition
>
> People expect unique quotients being all "isomorphic"
> to the complete ordered field, it isn't. Complex
> numbers _have_ other quotients, real numbers from
> the complete ordered field have _unique_ quotients.
>
> What's left after truncating a piece that exists
> fits, though it's kind of amputated. Like, when
> Cinderella's step-sister's slipper fit after
> she cut her toes off to fit the slipper.
>
> That any complex-number, has, n-ary roots, ...
> Well any number has n-ary roots.
>
> I think you mean "unity has n'th complex roots".
>
> There's the fundamental theorem of algebra, ...,
> that that says a polynomial of n'th order has n many roots,
> that though the multiplicity of roots isn't necessarily 1.
>
> It's so though that positive real numbers
> have unique positive real roots.
>
>
> How about "roots of phi", ..., powers of phi are
> pretty directly figured, yet, roots, ....
>
>
> The, "roots of zero" then is about where it is so
> that for some integral equations, it would be, an,
> indeterminate quantity, at zero, yet it's still
> part of the domain, so, something like zero is
> part of the "envelope", of the linear fractional
> equation, and Clairaut's equation, and d'Alembert's equation,
> and so is x = y = z = ..., "the identity dimension",
> an "origin".
>
>

"Roots of Identity"