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NNTP-Posting-Date: Sun, 17 Nov 2024 04:18:33 +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: Sat, 16 Nov 2024 20:18:42 -0800
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On 11/16/2024 07:58 PM, Chris M. Thomasson wrote:
> 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).

Those are usually called "2-tuples", members of R^2.

Then, relating those real numbers to complex numbers
is usually called "the Argand diagram" or "the Wessel diagram",
then the idea that functions accept complex or real valued
variables, is according to being "analytic", then as with
regards to whether under transformations what results "real analytic",
sort of whether in the z-order of R^2 and C their diagram
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