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

On 11/11/2024 1:00 AM, WM wrote:

On 10.11.2024 21:36, Chris M. Thomasson wrote:

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.

 But intervals on the real axis can be translated.

>
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!?

 There it is easier to contradict Cantor, because naturals and rationals can be interpreted as points on the real axis.
If pairing of naturals and rationals is possible for the complete set, then all intervals with natural numbers as midpoints
I(n) = [n - 1/10, n + 1/10]
can be translated until all rational numbers
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1,  ...
are midpoints.
 Obviously that is impossible because the density 1/5 of the intervals can never increase. It is possible however to shift an arbitrarily large (a potentially infinite) number of intervals to rational midpoints.

I don't think you know how to take any natural number and turn it into a unique pair, and then back again via Cantor pairing.