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

On 11/10/2024 1:35 AM, WM wrote:

On 10.11.2024 00:27, Jim Burns wrote:

On 11/9/2024 6:45 AM, WM wrote:

 

Everybody who believes that the intervals
I(n) = [n - 1/10, n + 1/10]
could grow in length or number
to cover the whole real axis
is a fool or worse.

>
Our sets do not change.
>
The set
   {[n-⅒,n+⅒]: n∈ℕ⁺}
with the midpoints at
   ⟨ 1, 2, 3, 4, 5, ... ⟩
does not _change_ to the set
   {[iₙ/jₙ-⅒,iₙ/jₙ+⅒]: n∈ℕ⁺}
with the midpoints at
   ⟨ 1/1, 1/2, 2/1, 1/3, 2/2, ... ⟩

 It cannot do so because the reality of the rationals is much larger than the reality of the naturals.[...]

Cantor pairing can create a unique pair of natural numbers from a single natural number. Why do think of rationals at all!? Sigh.