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

On 18.01.2025 00:08, Jim Burns wrote:

On 1/17/2025 2:40 PM, WM wrote:

Am 17.01.2025 um 17:53 schrieb Jim Burns:

On 1/17/2025 4:08 AM, WM wrote:

On 16.01.2025 23:22, Jim Burns wrote:

 

Nowhere,
among what appears and
among what doesn't appear,
is there finite ω-1 and infinite (ω-1)+1

>
So it appears because ω and ω-1 are dark.

>
We never see ω and ω-1
We see descriptions of ω and ω-1
That is sufficient for knowledge of ω and ω-1

>
Dark numbers cannot be seen,
if you understand by that phrase
be put in a FISON.

 Definitions can be seen.

Yes, dark numbers however can be handled only collectively. That distinguishes them from visible numbers.

 Finite sequences of claims, each claim of which
is true.or.not.first.false
can be seen.

Like the visible numbers.

 ----
The finite extends
much further than you (WM) think it does.
Infinitely further than you think it does.

No. As long as you deny Bob's existence and violate logic you are not a reliable source.

No finite ordinal has
an infinite immediate successor.

Maybe. But then there is no infinite ordinal.

You (WM) introduce
negative cardinality (darkᵂᴹ numbers)
in an attempt to fit these claims together.

>
No, I don't.

 I'm willing to believe that
you didn't intend to introduce negative cardinality.
Nonetheless, you did.
 A potentiallyᵂᴹ infinite set is larger.than.any.finite.

No, it is a finite step in a process with no upper bound.

 An actuallyᵂᴹ infinite set A isn't potentially infinite.
It isn't larger.than.any.finite.

It is much larger. Every finite is set is infinitesimal compared to ω, namely it is smaller than ω/n for every visible number n.

There is a larger finite set F.

No.

 Actuallyᵂᴹ infinite A has
a potentiallyᵂᴹ infinite subset P

Yes, but better this is expressed as subcollection or a multitude of finite sets.
Regards, WM