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

On 12/19/2024 9:50 AM, WM wrote:

On 18.12.2024 21:21, Jim Burns wrote:

On 12/18/2024 2:14 PM, WM wrote:

On 18.12.2024 19:22, Jim Burns wrote:

On 12/18/2024 7:40 AM, WM wrote:

On 18.12.2024 02:16, Jim Burns wrote:

The infinite end.segments of finite.cardinals
do not include any finite end.segments and
they have an empty intersection.

>
Explicitly wrong.
As long as
only infinite endsegments are concerned
their intersection is infinite.

f(k) = ∩{E(1), E(2), ..., E(k)}
with E(1) = ℕ
∀k ∈ ℕ :
∩{E(1), E(2), ..., E(k+1)} =
∩{E(1), E(2), ..., E(k)} \ {k},
∩{E(1), E(2), ...} is empty.
Other sets are not in the argument.

>
∀k ∈ ℕ :
k ∉ E(k+1) ⊇ ⋂{E(1),E(2),...} ∌ k
⋂{E(1),E(2),...} is empty.
Other numbers are not in the argument.

>
But E(1) is full.
The only way to get rid of content is
to proceed by
∀k ∈ ℕ:
∩{E(1), E(2), ..., E(k+1)} =
∩{E(1), E(2), ..., E(k)} \ {k},
i.e. to lose one number per term of
the function
f(k) = ∩{E(1), E(2), ..., E(k)}.

f(k) = E(k) = ⟦k,ℵ₀⦆
Yes,
after infinitely.many one.number.losses,
infinitely.many numbers are lost.
However,
each number is not.after infinitely.many numbers,
each loss is not.after infinitely.many losses.

The empty term

There is no darkᵂᴹ or visibleᵂᴹ empty term.
⎛ Assume otherwise.
⎜ Assume an empty term exists,
⎜ E(𝔊)\{𝔊} = {}
⎜ ⟦𝔊,ℵ₀⦆\{𝔊} = {}
⎜
⎜ ⟦0,ℵ₀⦆ is the set of cardinals of
⎜ sets smaller.than fuller.by.one sets.
⎜ b ∉ A  ⇒
⎜ #A < #(A∪{b})  ⇒  #A ∈ ⟦0,ℵ₀⦆
⎜
⎜ For each set smaller.than fuller.by.one sets,
⎜ fuller.by.ONE sets are smaller.than
⎜ fuller.by.TWO sets.
⎜ (b ∉ A ∌ c ≠ b)  ⇒
⎜ #A < #(A∪{b})  ⇒  #(A∪{b}) < #(A∪{b,c})
⎜
⎜ #(A∪{b}) < #(A∪{b,c})  ⇒  #(A∪{b}) ∈ ⟦0,ℵ₀⦆
⎜
⎜ For each cardinal k of
⎜ a set smaller.than fuller.by.one sets,
⎜ k+1 is also a cardinal of
⎜ a set smaller.than fuller.by.one sets.
⎜ k ∈ ⟦0,ℵ₀⦆  ⇒  k+1 ∈ ⟦0,ℵ₀⦆
⎜
⎜ Consider ⟦𝔊,ℵ₀⦆
⎜ 𝔊 ∈ ⟦0,ℵ₀⦆
⎜ 𝔊+1 ∈ ⟦0,ℵ₀⦆
⎜ 𝔊 ∈ ⟦𝔊,ℵ₀⦆
⎜ 𝔊 < 𝔊+1
⎜ 𝔊+1 ∈ ⟦𝔊,ℵ₀⦆
⎜ 𝔊+1 ∈ ⟦𝔊,ℵ₀⦆\{𝔊}
⎜ ⟦𝔊,ℵ₀⦆\{𝔊} ≠ {}
⎜
⎜ However,
⎜ ⟦𝔊,ℵ₀⦆\{𝔊} = {}
⎝ Contradiction.
Therefore,
there is no darkᵂᴹ or visibleᵂᴹ empty term.

The empty term
has lost all natnumbers one by one.

One.by.one and (darkᵂᴹ or visibleᵂᴹ) endlessly.
For each set smaller.than fuller.by.one sets,
fuller.by.ONE sets are smaller.than
fuller.by.TWO sets.
No set is largest among
the sets smaller.than fuller.by.one sets.