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

On 24.11.2024 21:17, Jim Burns wrote:

On 11/24/2024 2:42 PM, WM wrote:

On 24.11.2024 20:26, Jim Burns wrote:

 

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.

 Finite cardinalities can change by 1.

Endsegmentes can change by 1 element. Therefore their number of elements can change by 1.

 Infinite cardinalities are larger than
each finite cardinality,
and cannot change by 1.

Infinite cardinalities are to coarse to indicate that change. But the change takes place:
∀k ∈ ℕ : E(k+1) = E(k) \ {k}
==> ∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1

ℕ is the set of each and only finite cardinalities.
 |ℕ| isn't a finite cardinality.
|ℕ| cannot change by 1.

ℕ cannot change by 1 Element.
==> |ℕ| can change by 1.

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

 Cardinalities which cannot change by 1
do not change by 1 when they're called unserious.

 From two examples above you can see a change of an infinite set by 1 element. A proper measure will be able to indicate that.
Why do you wish to adhere to such an incapable measure?
Regards, WM