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

On 11/12/2024 4:38 PM, WM wrote:

On 12.11.2024 20:01, Jim Burns wrote:

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.

Are there more, though?
Or are there fewer?
i/j ↦ (i+j-1)(i+j-1)+2⋅i
⟨ 1/1 1/2 2/1 1/3 2/2 3/1 1/4 2/3 ... ⟩
↦
⟨ 2   4   6   8   10  12  14  16 ... ⟩
Or do infinite sets have different rules
than finite sets do?