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

On 11/23/2024 3:54 AM, WM wrote:

On 22.11.2024 23:56, Jim Burns wrote:

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

[...]

[...]

∀k ∈ ℕ_def:
∩{E(1), E(2), ..., E(k)} = E(k),
|E(k)| = ℵ₀

For all endsegments:
∀k ∈ ℕ:
|E(k+1)| = |E(k)| - 1

You (WM) mostly don't disagree with yourself
  in the same post.
Perhaps what you intend to say  is
⎛ For all post.definable end.segments
⎜ ∀k ∈ (ℕ\ℕ_def):
⎝ |E(k+1)| = |E(k)| - 1
Perhaps what you intend to say  is
⎛ There are post.definable end.segments
⎝ which have finite cardinalities.
I nearly agreed with that,
because our ℕ = ℕ_def
  so that says nothing,
  we have no such k
However,
your mention of 'E(k+1)' implies
⎛ ∀k ∈ (ℕ\ℕ_def):
⎝ (ℕ\ℕ_def) ∋ k+1
For _your_ end.segments,
k ↦ k+1 : one.to.one
E(k) → E(k+1) : one.to.one
|E(k)| ≤ |E(k+1)|
Also,
E(k) ⊇ E(k+1)
|E(k)| ≥ |E(k+1)|
|E(k)| = |E(k+1)|
Thus,
actually,
even your post.definable end.segments have
cardinalities which don't change by 1
  and thus are infinite cardinalities.

Therefore there must
exist infinitely many finite endsegments.

Your rabbit.out.of.the.hat:
more end segments than numbers in them.
----

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.

>
∀k ∈ ℕ_def:
∩{E(1), E(2), ..., E(k)} = E(k),
|E(k)| = ℵ₀
>
∩{E(1), E(2), ...} = { }.

⎛ ℕ_def  is
⎜ ℕ_def ∋ 0 ∧ ∀k ∈ ℕ_def: ℕ_def ∋ k+1
⎜ ℕ_def ⊆ S  ⇐  S ∋ 0 ∧ ∀k ∈ S: S ∋ k+1
⎜
⎜ ∀k ∈ ℕ_def:
⎜ |ℕ_def\E(k)| < |ℕ_def\E(k+1)| < |ℕ_def| = ℵ₀
⎜
⎝ Each end.segment of ℕ_def is countable.to.
⋂{E(k):k∈ℕ_def} = {}
because
∀j ∈ ℕ_def:
j ∉ E(j+1) ∈ {E(k):k∈ℕ_def}
j ∉ ⋂{E(k):k∈ℕ_def}
The end.segment.intersection is empty because
each end.segment of ℕ_def is countable.past.