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

On 24.11.2024 22:33, Jim Burns wrote:

On 11/24/2024 3:56 PM, WM wrote:

Endsegments can change by 1 element.
Therefore their number of elements can change by 1.

 Yes,
each end.segment.set can change by 1 element
 However,
  for each end.segment.set E(k)
  for each finite cardinality j
there is a larger.than.j subset E(k)\E(k+j+1)

Finite cardinalities belong to dark endsegments. Not every dark endsegment has a successor.

  For each end.segment.set E(k)
  for each finite cardinality j
j is not the cardinality of E(k)

The endsegments only can have an empty intersection if there are endsegments with 3, 2, 1, 0 elements.
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀
∀k ∈ ℕ : E(k+1) = E(k) \ {k}
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
∩{E(k) : k ∈ ℕ} = { }
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1
Regards, WM