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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
 (doubling-spaces)
Date: Sat, 16 Nov 2024 19:58:34 -0800
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On 11/12/2024 8:26 PM, Ross Finlayson wrote:
> On 11/12/2024 07:36 PM, Chris M. Thomasson wrote:
>> On 11/12/2024 6:45 PM, Ross Finlayson wrote:
>>> 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"
>>>
>>
>> n-ary roots a complex number a such that any of the roots when raised
>> back up by a power, say, n. equal the exact same complex number a. It's
>> really fun. Actually, it's hyper fun, read all if you get the time:
>>
>> https://paulbourke.org/fractals/multijulia/
>>
>> A friend of mine did a little write up on some of my work.
> 
> Yeah you posted this before and I commented about it then.
> 
> So, 1 + i0 ?
> 
> 

(1 + i0) complex number should be 2-ary point (1, 0).