Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 11.12.2024 23:28, Jim Burns wrote:

On 12/11/2024 4:53 PM, WM wrote:
 

Is the complete removal of natural numbers from the sequence of intersections
bound by the law
∀k ∈ ℕ :
∩{E(1), E(2), ..., E(k+1)} =
∩{E(1), E(2), ..., E(k)} \ {k}
or not?

 Each "leaves" by
not.being.in.common.with.all.end.segments.

That means no natural number remains in all endsegments. But every endsegment has only one number less than its predecessor. This closes the bridge between infinite intersections of endsegments and the empty intersection of endsegments.

 ∀k ∈ ℕ:
k ∉ E(k)\{k} = E(k+1) ⊇ ⋂{E(i):i} ∌ k
 ⋂{E(i):i} = {}.

There must be a continuous sequence of steps of height 1 from many elements to none. Can you confirm this?
Regards, WM