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

On 11/25/2024 8:52 AM, WM wrote:

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.

Finite cardinals can change by 1
Each end.segment Eᶠⁱⁿ(k) of the finite.cardinalities ℕᶠⁱⁿ
holds a countable.to.from.0 least.element
which is to say,
is not dark.
  For each end.segment.set Eᶠⁱⁿ(k) of ℕᶠⁱⁿ
  for each finite cardinality j
there is a larger.than.j subset E(k)ᶠⁱⁿ\Eᶠⁱⁿ(k+j+1)
j < j+1 = |Eᶠⁱⁿ(k)\Eᶠⁱⁿ(k+j+1)|
  For each end.segment.set Eᶠⁱⁿ(k)
  for each finite cardinality j
j is not the cardinality of Eᶠⁱⁿ(k)
j < |Eᶠⁱⁿ(k)\Eᶠⁱⁿ(k+j+1)| ≤ |Eᶠⁱⁿ(k)|
j ≠ |E(k)|
That contradicts |Eᶠⁱⁿ(k)| being any finite cardinal j
  any cardinal j which can change by 1
Which contradicts cardinality |Eᶠⁱⁿ(k)| changing by 1
even though set Eᶠⁱⁿ(k) changes by 1.

Not every dark endsegment has a successor.

Each dark end.segment is not
an end.segment of the finite.cardinals.

  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.

The end.segments
can only have a non.empty intersection
if there is an element which is in each end.segment.