Re: Incompleteness of Cantor's enumeration of the rational numbers

On 12.11.2024 20:01, Jim Burns wrote:

On 11/12/2024 11:43 AM, WM wrote:

No, the intervals remain constant
in size and multitude.

 Intervals which are constant _only_
in size and multitude
are not constant absolutely.

They would suffer to cover all rationals completely if Cantor's bijection was complete.

These intervals
{[n-⅒,n+⅒]: n∈ℕ⁺}
cover all naturals ℕ⁺  and
do not cover all fractions ℕ⁺/ℕ⁺

Right.

 

But the rationals are more in the sense that
they include all naturals and 1/2.

 These intervals
{[i/j-⅒,i/j+⅒]: i/j∈ℕ⁺/ℕ⁺}
cover all fractions ℕ⁺/ℕ⁺

But these are more intervals.

 These intervals
{[n-⅒,n+⅒]: n∈ℕ⁺}
and these intervals
{[i/j-⅒,i/j+⅒]: i/j∈ℕ⁺/ℕ⁺}
are different intervals.

In particular the second kind of intervals must be more. And this is the solution: The identity of the intervals for the geometric covering is irrelevant. I will elaborate o this in the next posting.
Regards, WM