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

On 13.11.2024 10:08, Jim Burns wrote:

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

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
 > ⟨ 1/1 1/2 2/1 1/3 2/2 3/1 1/4 2/3 ... ⟩
 > ↦
 > ⟨ 2   3   5   7   11  13  17  19 ... ⟩

 

or
 > ⟨ 1/1        1/2       2/1  ... ⟩
 > ↦
 > ⟨ 10^10   10^10^10  10^10^10^10  ... ⟩

Or do infinite sets have different rules
than finite sets do?

If infinite sets obey the rules sketched above, then set theorists must discard geometry
because by shifting intervals the relative covering 1/5 of ℝ+ becomes oo*ℝ,
and analysis
because the constant sequence 1/5, 1/5, 1/5, ... has limit oo,
and logic
because of Bob.
Regards, WM