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

On 1/13/2025 12:17 PM, WM wrote:

On 13.01.2025 17:33, Jim Burns wrote:

On 1/12/2025 2:39 PM, WM wrote:

On 12.01.2025 20:33, Jim Burns wrote:

On 1/12/2025 10:54 AM, WM wrote:

No, it depends on completeness.

>
It is completely true
that each natural number is a natural number   and
that only natural numbers are natural numbers.

>
and that nothing fits between them and ω.

>
Yes.

>
Therefore
doubling of all natural numbers
creates numbers larger than ω.

No.
No infinitesᵒʳᵈ are needed for
arithmetic of finitesᵒʳᵈ.

ω is first infiniteᵒʳᵈ.
No infiniteᵒʳᵈ is before ω
No finiteᵒʳᵈ is after ω

>
Right.

Remember:
No infinitesᵒʳᵈ are needed for
arithmetic of finitesᵒʳᵈ.

Regular distances in ⦅0,ω⦆ multiplied by 2
remain regular distances in ⦅0,ω⦆, not.in ⟦ω,2ω⦆

>
Doubling of all n
deletes the odd numbers
but cannot change the number of numbers,

ℕ is the set of finite ordinals.
There is no finite set larger than ℕ
thus ℕ is infinite.
There is no infinite set smaller than ℕ
𝔼 is the set of even finite ordinals.
There is no finite set larger than 𝔼
thus 𝔼 is infinite
𝔼 ⊆ ℕ
#𝔼 ≤ #ℕ
There is no infinite set smaller than ℕ
#𝔼 ≥ #ℕ
#𝔼 = #ℕ

therefore creates even numbers.
They do not fit below ω.

No.
They fit below ω
A step is never from finite to infinite.
Therefore, a step never crosses ω
Therefore, a sum never crosses ω
Therefore, a product never crosses ω
Therefore, a power never crosses ω

Remember:
nothing fits between them and ω.

Remember:
No infinitesᵒʳᵈ are needed for
arithmetic of finitesᵒʳᵈ.
#⟦0,k⦆ > #⟦1,k⦆  ⇒  k < ω
#⟦0,ζ⦆ = #⟦1,ζ⦆  ⇒  ω ≤ ζ
¬( #⟦0,k⦆ > #⟦1,k⦆  ∧  #⟦0,k+1⦆ = #⟦1,k+1⦆ )
¬( k < ω  ∧  ω ≤ k+1 )
¬( j+(k₀-1) < ω  ∧  ω ≤ (j+(k₀-1))+1 )
¬( j+(k₀-1) < ω  ∧  ω ≤ j+k₀ )
Well.order.
¬( j,k < ω  ∧  ω ≤ j+k )
¬( j×(k₀-1) < ω  ∧  ω ≤ j+(j×(k₀-1)) )
¬( j×(k₀-1) < ω  ∧  ω ≤ j×k₀ )
Well.order.
¬( j,k < ω  ∧  ω ≤ j×k )
¬( j^(k₀-1) < ω  ∧  ω ≤ j×(j^(k₀-1)) )
¬( j^(k₀-1) < ω  ∧  ω ≤ j^k₀ )
Well.order.
¬( j,k < ω  ∧  ω ≤ j^k )