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

On 1/18/2025 3:41 AM, WM wrote:

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.

Definitions of visibleᵂᴹ.or.darkᵂᴹ numbers
are true claims about visibleᵂᴹ.or.darkᵂᴹ numbers
and do not distinguish them.
For example,
visibleᵂᴹ.or.darkᵂᴹ ordinals are defined to be
well.ordered:
Each non.empty set of them holds
a visibleᵂᴹ.or.darkᵂᴹ first in that set.
A true claim.
A true.or.not.first.false claim.
Ask me for a useful fact about finite sequences of
true.or.not.first.false claims.

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

>
Like the visible numbers.

Not imagined.to.be.visibleᵂᴹ.
Visible.
On paper. In chalk. Or in glowing dots.
Carved into marble. Or into clay tablets.
Including.but.not.limited.to
claims about and not.distinguishing.between
visibleᵂᴹ.or.darkᵂᴹ ordinals.
Including.but.not.limited.to
⎛
⎜ Visibleᵂᴹ.or.darkᵂᴹ ordinals are
⎜ well.ordered.
⎜
⎜ Set {j,k} of visibleᵂᴹ.or.darkᵂᴹ ordinals
⎜ holds a first visibleᵂᴹ.or.darkᵂᴹ ordinal.
⎜
⎜ j ≠ k  ⇒  j < k  ∨  j > k
⎜
⎜ Set {i,j,k} of visibleᵂᴹ.or.darkᵂᴹ ordinals
⎜ holds a first visibleᵂᴹ.or.darkᵂᴹ ordinal.
⎜
⎜ i < j  ∧  j < k  ⇒  i < k
⎜
⎜ ¬(k < k)
⎜
⎜ Visibleᵂᴹ.or.darkᵂᴹ ordinals are
⎜ linearly ordered.
⎜
⎜ Darkᵂᴹ ordinals are linearly ordered
⎜ with respect to visibleᵂᴹ ordinals and
⎝ with respect to other darkᵂᴹ ordinals.
Somewhere Out There, there might be
a person thinking that
"This is darkᵂᴹ"
implies
"This is something we must remain ignorant of".
A person thinking that would be wrong.