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

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.
There are no limits involved.

>
Then it would be great if you (WM)
stopped calling things 'limit sets'

>
The sets described here are not limit sets.

 Q. What is a limit set?
 

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.

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)}.
The empty term has lost all natnumbers one by one.
Regards, WM