Re: Incompleteness of Cantor's enumeration of the rational numbers (doubling-spaces)

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