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NNTP-Posting-Date: Wed, 13 Nov 2024 04:26:23 +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 20:26:19 -0800
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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 ?