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

On 05.01.2025 20:54, Jim Burns wrote:

On 1/5/2025 6:07 AM, WM wrote:

On 04.01.2025 17:20, Jim Burns wrote:

On 1/4/2025 3:42 AM, WM wrote:

On 1/3/2025 3:56 PM, Jim Burns wrote:

 

All finite.ordinals removed from
the set of each and only finite.ordinals
leaves the empty set.

>
But removing
every ordinal that you can define
(and all its predecessors) from ℕ leaves
almost all ordinals in ℕ.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

>
ℕ is the set of each and only finite.ordinals.

>
Yes.

 Anything which is a finite.ordinal,
darkᵂᴹ and visibleᵂᴹ, is in ℕ

Yes!

Q. What is a finite.ordinal?

Every ordinal smaller than ω = |ℕ| is a finite ordinal. 1, 2, 3, ..., ω/1000000, ..., ω/2, ω-1 are finite ordinals.

⎛ It is an element of ℕ
⎜
⎜ It is an ordinal,
⎜ one of the (well.ordered) ordinals.
⎜
⎜ It is
⎜ larger.by.one than emptier.by.one
⎜  ( #⟦0,k⦆ > #(⟦0,k⦆\⦃0⦄)
⎜ or it is emptiest.
⎝  ( k = 0 = ⦃⦄

0 is a cardinal but not an ordinal. Who would be the zeroest?

 We know that
all of that is true.without.exception
among the finite.ordinals,
whether.or.not I can giveᵂᴹ the finite.ordinal,
whether.or.not it's darkᵂᴹ.

Yes.

 

The sequence of end.segments of ℕ
grows emptier.one.by.one but
it doesn't grow smaller.one.by.one.

>
It does
but you cannot give the numbers
because they are dark.

 Not.giving numbers doesn't prevent us from
making claims which we know are without.exception.

Yes. Therefore the dark natural numbers should have the same properties as the visible numbers . except one.

 

A precise measure must detect
the loss of one element.
ℵo is no precise measure but only
another expression for infinitely many.

 An accurate measure must recognize that
a set larger.than any.finite.set
is not any.finite.set.
 https://en.wikipedia.org/wiki/Accuracy_and_precision
 For each finite.set, there is
a finite.ordinal of the same size.
 For each finite.ordinal, there is
a larger.by.one finite.ordinal,
and
it and its priors are a subset of
the set of all finite.ordinals.
 For each finite.set, there is
a larger.than.that subset of
the set of all finite.ordinals.
 For each finite.set, that set is not
the set of all finite.ordinals.
 The set of all finite.ordinals is not
any.finite.set.
 Q. What is a finite set?
 

What is a set that can be recognized as finite? It has a visible n as its cardinality. But the set {1, 2, 3, ..., ω/2} is also finite because |{1, 2, 3, ..., ω/2}| = ω/2 < ω but cannot be recognized as finite because we cannot count to any fraction of ω. It appears like an infinite set which it is not by definition.
Regards, WM