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

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.
Finite sequences of claims, each claim of which
is true.or.not.first.false
can be seen.
----
The finite extends
much further than you (WM) think it does.
Infinitely further than you think it does.
A non.0 ordinal k for which
it and each of its non.0 priors j < k
have immediate predecessors k-1 and j-1
is
a finite ordinal.
No finite ordinal has
an infinite immediate successor.
It becomes clear why,
if we go to the definition.
If k is finite, then
each in (0,k] has an immediate predecessor.
If k+1 is finite, then
each in (0,k+1] has an immediate predecessor.
(0,k+1] = (0,k]∪{k+1}
k+1 has an immediate predecessor k
and
everything else in (0,k+1] (that is, in (0,k])
has an immediate predecessor
if k is finite.

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.
An actuallyᵂᴹ infinite set A isn't potentially infinite.
It isn't larger.than.any.finite.
There is a larger finite set F.
Actuallyᵂᴹ infinite A has
a potentiallyᵂᴹ infinite subset P
larger.than.any.finite, larger than F, specifically.
#P > #F > #A
A ⊇ P