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

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