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

On 12/29/2024 5:34 AM, WM wrote:

On 28.12.2024 18:31, Jim Burns wrote:

On 12/28/2024 9:03 AM, WM wrote:

[0, ω-1] = [0,ω⦆ = ℕ =/= [0, ω]

>
Yes,
⟦0,ω⦆  =  ℕ  ≠  ⟦0,ω⟧
>
However,
⎛ Assume k < ω
⎜ #⟦0,k⦆ ≠ #(⟦0,k⦆∪⦃k⦄)
⎜ #⟦0,k+1⦆ ≠ #(⟦0,k+1⦆∪⦃k+1⦄)
⎝ k+1 < ω

 That holds for almost all natural numbers k.

Most of the above is by definition.
k < ωⁿᵒᵗᐧᵂᴹ  :⇔  #⟦0,k⦆ ≠ #⟦0,k+1⦆
⟦0,k+1⦆ := ⟦0,k⦆∪⦃k⦄
However, you (WM) insist that
#⟦0,ωᵂᴹ⦆ ≠ #⟦0,ωᵂᴹ+1⦆
thus
ωᵂᴹ < ωⁿᵒᵗᐧᵂᴹ
ωᵂᴹ ∈ ⟦0,ωⁿᵒᵗᐧᵂᴹ⦆
And, because, for ωⁿᵒᵗᐧᵂᴹ in general,
⎛ ∀j ∈ ⟦0,ωⁿᵒᵗᐧᵂᴹ⦆:
⎝ (∀)k ∈ ⟦0,ωⁿᵒᵗᐧᵂᴹ⦆: k ∈ ⦅j,ωⁿᵒᵗᐧᵂᴹ⦆
  we also have
(∀)k ∈ ⟦0,ωⁿᵒᵗᐧᵂᴹ⦆: k ∈ ⦅ωᵂᴹ,ωⁿᵒᵗᐧᵂᴹ⦆
  which means
∃᳹k ∈ ⟦0,ωⁿᵒᵗᐧᵂᴹ⦆:  k ∈ ⦅ωᵂᴹ,ωⁿᵒᵗᐧᵂᴹ⦆
¬∃᳹k ∈ ⟦0,ωⁿᵒᵗᐧᵂᴹ⦆:  ¬(k ∈ ⦅ωᵂᴹ,ωⁿᵒᵗᐧᵂᴹ⦆)
  which means
#⦅ωᵂᴹ,ωⁿᵒᵗᐧᵂᴹ⦆  >ᵉᵃᶜʰ  ⦃#⟦0,i⦆: #⟦0,i⦆≠#⟦0,i+1⦆ ⦄
¬(#⟦0,ωᵂᴹ⦆  >ᵉᵃᶜʰ  ⦃#⟦0,i⦆: #⟦0,i⦆≠#⟦0,i+1⦆⦄ ⦄)
Almost.all of our finitesⁿᵒᵗᐧᵂᴹ ⦅ωᵂᴹ,ωⁿᵒᵗᐧᵂᴹ⦆
are not your finitesᵂᴹ.
----
The following is more than a definition.
One aspect, variously useful and frustrating,
of being not.a.definition is that
not.a.definition can't be defined out of being.true,
the way in which ωᵂᴹ conceivably "replaces" ωⁿᵒᵗᐧᵂᴹ.
⎛
⎜ Assume #⟦0,k⦆ ≠ #(⟦0,k⦆∪⦃k⦄)
⎜
⎝ #⟦0,k+1⦆ ≠ #(⟦0,k+1⦆∪⦃k+1⦄)
Which is to say:
no finite is the last of the finites
-- by virtue of a particular use of:
⎛
⎜ Assume #A ≠ #B
⎜
⎝ #(A∪{a}) ≠ #(B∪{b})
⎛ #A ≠ #B  ⇐
⎜ #A < #B  xor
⎜ #B ≤ #A  iff
⎝ ∃f one.to.one: B ⇉ A
#(B∪{b}) ≤ #(A∪{a})  ⇒  #B ≤ #A
⎛ Assume #(B∪{b}) ≤ #(A∪{a})
⎜
⎜ ∃g one.to.one: B∪{b} ⇉ A∪{a}
⎜⎛ g(b) = g(b)    [!]
⎜⎜ g(g⁻¹(a)) = a  [!]
⎜⎝ otherwise g(x) = g(x)
⎜
⎜ Define f one.to.one: B∪{b} ⇉ A∪{a}
⎜⎛ f(b) = a          [!]
⎜⎜ f(g⁻¹(a)) = g(b)  [!]
⎜⎝ otherwise f(x) = g(x)
⎜
⎜ ∃f one.to.one: B ⇉ A
⎜⎛ f(g⁻¹(a)) = g(b)
⎜⎝ otherwise f(x) = g(x)
⎜
⎝ #B ≤ #A
Therefore,
#(B∪{b}) ≤ #(A∪{a})  ⇒  #B ≤ #A
#A < #B  ⇒  #(A∪{a}) < #(B∪{b})
because [!]
swapping two values of a one.to.one map
leaves another one.to.one map.
No finite is the last of the finites
because
#A < #B  ⇒  #(A∪{a}) < #(B∪{b})

However,
⎛ Assume k < ω
⎜ #⟦0,k⦆ ≠ #(⟦0,k⦆∪⦃k⦄)
⎜ #⟦0,k+1⦆ ≠ #(⟦0,k+1⦆∪⦃k+1⦄)
⎝ k+1 < ω

>
That holds for almost all natural numbers k.
It cannot hold for an actually infinite system
without disappearing Bob.

Bob can disappear within a larger.enough set,
because
no finite is last of the finites,
because
swapping two values of a one.to.one map
leaves another one.to.one map.

It cannot hold for an actually infinite system
without disappearing Bob.

A potentiallyᵂᴹ infinite set 𝔸
cannot be completedᵂᴹ to actuallyᵂᴹ infinite 𝔸∪𝔻
by any epilogue 𝔻 such that ∀d ∈ 𝔻: g(d) = d