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

On 14.12.2024 19:57, Jim Burns wrote:

On 12/14/2024 5:26 AM, WM wrote:

Don't say what not is.

 For sets A and B with one.to.one A.to.B
B is not.smaller.than A

Don't say what not is.

Don't say what not is.

 If ω-1 exists
then
⎛ ω-1 is last.before.ω
⎜ ω is first bound of the finites
⎜ ω-1 is not any before.ω bound of the finites
⎜ ω-1 is not infinite (see [2])
⎜ ω-1 is finite
⎜ ω-1 is smaller.than (ω-1)+1, also finite (see [3])
⎜ (ω-1)+1 is smaller.than (ω-1)+2, also finite
⎜ ω bounds w-1, (ω-1)+1, (ω-1)+2
⎜ ω-1 < (ω-1)+1 < (ω-1)+2 ≤ ω
⎝ ω-1 is not last.before.ω

Don't say what not is. Explain your vision of the problem:
If ℕ is a set, i.e. if it is complete such that all numbers can be used for indexing sequences or in other mappings, then it can also be exhausted such that no element remains. Then the set of what remains unused, i.e., of intersections of endsegments
(1)   E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
loses all content. Then, by the law
(2)   ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
the content must become finite.
Regards, WM