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

On 24.11.2024 20:26, Jim Burns wrote:

On 11/24/2024 9:05 AM, WM wrote:

On 23.11.2024 23:10, Jim Burns wrote:

On 11/23/2024 4:39 PM, WM wrote:

On 23.11.2024 22:20, Jim Burns wrote:

 

Do you (WM) object to
  k ↦ k+1 : one.to.one

>
I don't know what that waffle should mean.

>
k ↦ k+1  means the successor operation.
>
'One.to.one' means that,
if j≠k  then j+1≠k+1
different numbers have different successors.
>
I am claiming that
different numbers have different successors.

>
Ok.

 "Ok, I understand you"
  or
"Ok, different numbers have different successors"
  ?

Yes.

What we mean by
  |E(k)| ≤ |E(k+1)|
is that
there is a one.to.one function
  from E(k) to E(k+1)
The successor operation, for example.

What I mean is the fact that
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1
whereas Cantor's ℵo is a very unsharp measure.

 So, there is.
So, |E(k)| ≤ |E(k+1)|  isn't wrong.

Cantor's nonsense has many faces. It i not suitable for serious maths.
Regards, WM