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NNTP-Posting-Date: Tue, 12 Nov 2024 23:13:33 +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: Tue, 12 Nov 2024 15:13:23 -0800
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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

2 + 2 = 2^2

1 + 1 =/= 1^2

1/phi = phi - 1

The doubling-spaces and doubling-measures are
a feature of quasi-invariant measure theory
with regards to the mathematical act of
individuation of a continuum.

With regards to the mathematical act: a mathematical fact.

Then, many would rather have that
they don't need your "pro-tection",
and that it gives higher thoughts short-shrift.

The densities of the integers and rationals,
or integers and squares,
or about the square-free numbers,
or about the primes,
have distinct expressions.

In the real values, ....


It's much appreciated a _usual_
"mathematical conscience", and a
thoroughly contemplative and deliberative
addressal of the import, and, limitations,
of plain "non-self-contradictory, inductive inference.

Then here the differences in "unboundedness" and
"density", do have that "density" remains defined,
including in the unbounded. Also it's a size relation.

Good luck and warm regards kind sirs