Re: Incompleteness of Cantor's enumeration of the rational numbers

On 17.11.2024 21:51, FromTheRafters wrote:

It happens that WM formulated :

On 17.11.2024 17:59, FromTheRafters wrote:

WM pretended :

On 17.11.2024 12:38, FromTheRafters wrote:

WM presented the following explanation :

On 17.11.2024 12:01, FromTheRafters wrote:

WM was thinking very hard :

On 16.11.2024 22:33, Moebius wrote:
>

For example "aleph_0 - aleph_0" is not defined.

>
Small wonder. ℵo means only infinitely many: |ℕ|, |ℚ|, and many others.
|ℕ|-|ℕ| however is defined.

>
No, it is not.

>
If sets are invariable then ℕ \ ℕ is empty.
If |ℕ| concerns only the elements of ℕ, then |ℕ|-|ℕ|= 0.

>
So, you're saying that if I take aleph_zero natural numbers and I remove the aleph_zero odd numbers from consideration in a new set, I will have a new emptyset instead of E?

>
Try to understand. "aleph_0 - aleph_0" is not defined.

>
Try to understand that |N| equals aleph_zero.

>
Of course. ℵo equals |ℕ|, equals |ℚ|, equals all countable sets. It is simply another name for infinitely "many". |ℕ| however is a fixed infinite number. Note that sets are invariable.

 But you said both that it equals zero and that it is undefined. You should pick one and be consistent.

I said that |ℕ| and |ℚ| and |ℕ| - 5 and |ℕ| + 6 etc. equal ℵo. That means that ℵo is nothing but "infinitely many". Therefore ℵo - ℵo is undefined but |ℕ| - |ℕ| = 0 and |ℚ| - |ℕ| > 0.
|ℕ| - |ℕ| = 0 because if you subtract one element from ℕ then you have no longer ℕ and therefore no longer |ℕ| describing it.
Regards, WM