Re: Incompleteness of Cantor's enumeration of the rational numbers (cardinals)

On 11/17/2024 08:59 AM, 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.

I'm wouldn't say from experience that it'll ever learn.
There's no such thing as subtraction in cardinals,
only ^ and <.
There's setminus w.r.t. sets: that though results what are that
sets have cardinals, being in an equivalence class
of Cartesian bijections with all other members of
the same cardinal an equivalence class as after the
Cantor-Schroeder-Bernstein theorem and Cartesian bijections
that making the equivalence relation, ..., sometimes
people confuse that with initial ordinals yet that's
considered wrong as ordinals and cardinals are different,
and a cardinal is an equivalence class of sets, though
it gets rather involved how big they are, when "1" the
cardinal is the largest equivalence class in ZF.
Then of course according to me, there are some
non-Cartesian functions like the natural/unit equivalency function,
or "EF", "sweep", not-a-real-function yet standardly modeled by
real functions, so that it can demonstrate for the analog/digital
and continuous/discrete a Jordan measure/content a line segment
geometrically as a continuous bounded domain, countable.
This of course is all framed up about "anti-diagonal" the
Cartesian and "only-diagonal" the non-Cartesian, since
the "anti-diagonal" sometimes sort of wrongly called
"The Diagonal Argument" is used for so many things
in ordinary algebras about set theory with cardinals.