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

On 06.01.2025 23:43, Jim Burns wrote:

On 1/5/2025 1:14 PM, WM wrote:

ℕ cannot be covered by FISONs,
neither by many nor by their union.
If ℕ could be covered by FISONs
then one would be sufficient.

 ℕ is the set of finite.ordinals.
ℕ holds each finite ordinal.
ℕ holds only finite.ordinals.

|ℕ| is by definition the smallest transfinite number, larger than every natural number by every visible factor.

 k ∈ ℕ  ⇒  k+1 ∈ ℕ
is true for both the darkᵂᴹ and the visibleᵂᴹ.

One exception exists: ω-1.
(0, ω)*2 = {2, 4, 6, ..., ω, ω+2, ω+4, ...}.

 

If ℕ could be covered by FISONs
then one would be sufficient.

 ℕ is the set of finite.ordinals.
 A FISON is a set of finite.ordinals
up to that FISON's maximum (finite.ordinal) element.
 If one FISON covered ℕ,
that FISON.cover would equal ℕ,
and the maximum of that FISON.cover
would be the maximum.of.all finite.ordinal.
 However,
no finite.ordinal k is the maximum.of.all.
k ∈ ℕ  ⇒  k+1 ∈ ℕ
That is true for both the darkᵂᴹ and the visibleᵂᴹ.

Not for all. There is an exception.

 Contradiction.
No one FISON covers ℕ.

No union of FISON covers ℕ.
  Every union of FISONs {1, 2, 3, ..., n} which stay below a certain threshold stays below that threshold. Every FISON stays below 1 % of ℕ because when extended by 100 {1, 2, 3, ..., 100n} is less than ℕ.
The FISONs form a potentially infinite collection whereas ℕ is an invariable set with |ℕ| is a fixed transfinite number larger than every finite number |{1, 2, 3, ..., n}| = n.

ℕ is the set of finite ordinals.
 Each finite.ordinal k is in
at least one FISON: ⟦0,k⟧
 Each finite.ordinal is in
the union of FISONs

Not the dark ones.

 The union of FISONs covers
the set ℕ of finite.ordinals

Contradiction. Should be recognizable to you.

 

But for all we have:
Extension by 100 is insufficient.

 Correct.
Which is weird, but accurate.

It is not weird but wrong.

 The source of that weird result is lemma 1.

No, the source it potential infinity.
Regards, WM