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

On 11/12/2024 1:06 PM, WM wrote:

On 12.11.2024 17:47, Jim Burns wrote:

[4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

>
|[4-⅒,4+⅒]| = |[1/3-⅒,1/3+⅒]|
and only that is important for my argument.

Yes,
μ[4-⅒,4+⅒] = μ[1/3-⅒,1/3+⅒]
⎛ Also, |[0,1]| = |[0,2]|
⎝ so I think you mean 'measure', not 'cardinality'.
The value of a measure is defined to be
in the extended reals≥0.
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }  ∈  [0,+∞]
The extended reals≥0 are
the Archimedean reals≥0 [0,+∞)
plus one non.Archimedean point +∞
[0,+∞] = [0,+∞)∪{+∞}
The value of μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }
is not any Archimedean point, that is,
there is no finite m ∈ ℕ⁺ such that
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ } ≤ m
The value of μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }
is not Archimedean, so must be +∞
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }  =  +∞
Also,
μ⋃{ [i/j-⅒,i/j+⅒]:i/j∈ℕ⁺/ℕ⁺ }  =  +∞

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.

>
Then you will get my point, hopefully.

Your point is that
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }
isn't in the extended reals.
I get it.
Getting it and
agreeing with your changes to our definitions,
changes which turn the discussion into gibberish,
are different.