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

On 11/22/2024 4:30 PM, WM wrote:

On 22.11.2024 19:51, Jim Burns wrote:

On 11/21/2024 4:57 PM, WM wrote:

On 11/22/2024 1:51 PM, Jim Burns wrote:

On 11/21/2024 4:57 PM, WM wrote:

On 21.11.2024 22:46, Jim Burns wrote:

The _description_ is completed.
It's right there.

>
The description of the set
not of all its elements.

>
The description

⎛ (i+j) := ⌈(2⋅n+¼)¹ᐟ²+½⌉
⎝ (i+j-1)⋅(i+j-2)/2+i = n

is sufficient in order to
finitely.investigate infinitely.many.

>
For instance,
not all indices of the endsegments
can be counted to.

You aren't referring to
the set ℕ⁺ of countable.to from.1
For n,i,j countable.to from.1
n ↦ ⟨i,j⟩ ↦ n: one.to.one
ℕ⁺ covers ℕ⁺×ℕ⁺  and  ℕ⁺×ℕ⁺ covers ℕ⁺
ℕ⁺ and ℕ⁺×ℕ⁺ are infinite sets.

Remember:
The intersection of all endsegments is empty,
but the intersection of
endsegments which can be counted to
is infinite.

No one should "remember" that.
It is incorrect.

Note that every endsegment loses only one number.

Finite cardinalities can lose one number.
After all finite cardinalites,
there are the infinite cardinalities,
which aren't finite,
and can't lose one number.

Therefore there must
exist infinitely many finite endsegments.

In ℕ⁺
each foresegment is finite.
If any endsegment is finite,
the union of endsegment and its foresegment is finite.
The union of any endsegment and its foresegment
is ℕ⁺
ℕ⁺ isn't finite,
and no endsegment is finite.