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

On 11/11/2024 11:00 AM, Ross Finlayson wrote:

On 11/11/2024 10:38 AM, Jim Burns wrote:

On 11/11/2024 4:00 AM, WM wrote:

On 10.11.2024 21:36, Chris M. Thomasson wrote:

On 11/10/2024 1:35 AM, WM wrote:

On 10.11.2024 00:27, Jim Burns wrote:

On 11/9/2024 6:45 AM, WM wrote:

>

Everybody who believes that the intervals
I(n) = [n - 1/10, n + 1/10]
could grow in length or number
to cover the whole real axis
is a fool or worse.

>
Our sets do not change.

>
But intervals on the real axis can be translated.

>
The interval [4-⅒,4+⅒] can be translated to
the interval [1/3-⅒,1/3+⅒].
>
[4-⅒,4+⅒] does not change to [1/3-⅒,1/3+⅒]
[4-⅒,4+⅒] will continue being after translation.
[1/3-⅒,1/3+⅒] has never been [4-⅒,4+⅒].
>
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?
>

(We had a great long thread over on sci.logic
about Banach-Tarski and Vitali-Hausdorff, there's
quite a bit about the historical and technical arrival,
including references and links to Hausdorff's original.
Vitali's doubling-space reflects on "Zeno's graduation
course", where Zeno also has a doubling-space or
doubling-measure argument, since about 2300 years ago.
These are considered part of "mathematics", if
your project is wider than "bumbing- or dumbing-down W.M.".)