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

On 11/12/2024 9:10 AM, WM wrote:

On 12.11.2024 05:32, Jim Burns wrote:

On 11/11/2024 3:40 PM, WM wrote:

On 11.11.2024 21:09, Jim Burns wrote:

On 11/11/2024 2:04 PM, Ross Finlayson wrote:

How about Banach-Tarski equi-decomposability?

>
The parts do not change.

>
Neither do my intervals [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒].

>
When I first read that,
I thought you meant [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒]
Later,
I thought you meant [4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

>
Both intervals are one and the same,
only shifted a bit.

No.
[4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]
⎛ Assume otherwise.
⎜ 4 ∈ [4-⅒,4+⅒]
⎜ Translateᵂᴹ.
⎜ 4 ∉ [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒]
⎜
⎜ ’Twas brillig, and the slithy toves
⎜  Did gyre and gimble in the wabe:
⎜ All mimsy were the borogoves,
⎝  And the mome raths outgrabe.
The borogove.claims are not.first.false.
It doesn't matter where your interval is,
one of those prior claims is false.
But
we can't accept mimsy borogroves and slithy toves.
They're nonsense, of course.
We must surrender our telescope into infinity:
finite sequences of claims which are
only true.or.not.first.false claims.
One can't have both
transmogrifying intervals  and
finite sequences giving knowledge of infinity.

Or is it by accident that
you used n = 4 to cover q = 1/3?
1/1, 1/2, 2/1, 1/3, ...

No accident.
You got my point.