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

On 12/31/2024 1:20 PM, WM wrote:

On 31.12.2024 18:25, Jim Burns wrote:

<WM>

Every FISON is less than 1 % of ℕ
because
by expanding it by a factor 100
the situations remains the same - forever.

</WM>
Date: Tue, 31 Dec 2024 10:39:12 +0100
Message-ID: <vl0e3v$25vs8$1@dont-email.me>
>
For each ordinal k, there is
an ordinal k+1 which is fuller.by.one.
⎛ k = ⟦0,k⦆
⎝ k+1 = ⟦0,k⦆∪⦃k⦄
>
For each finite.ordinal k, there is
a finite.ordinal k+1 which is larger.by.one.
⎛ finite k
⎜ #⟦0,k⦆ < #⟦0,k+1⦆
⎜ ¬∃f one.to.one: ⟦0,k+1⦆ ⇉ ⟦0,k⦆
⎜ ¬∃g one.to.one: ⟦0,k+2⦆ ⇉ ⟦0,k+1⦆
⎜ #⟦0,k+1⦆ < #⟦0,k+2⦆
⎝ finite k+1

>

For each room k, there is
swap k⇄k+1

>
That does not disprove the theorem:
Every union of FISONs which
stay below a certain threshold
stays below that threshold.
Note: Every FISON stays below 1 % of ℕ.

For each finite ⟦0,j⦆, there are
more.than.#⟦0,j⦆.many finite ⟦0,k⦆
#{⟦0,i⦆:#⟦0,i⦆<#⟦0,i+1⦆} >ᵉᵃᶜʰ {#⟦0,i⦆:#⟦0,i⦆<#⟦0,i+1⦆}
The set {⟦0,i⦆:#⟦0,i⦆<#⟦0,i+1⦆} of
all the finite ⟦0,k⦆
is not a finite set.
¬(#{⟦0,i⦆:#⟦0,i⦆<#⟦0,i+1⦆} ∈ {#⟦0,i⦆:#⟦0,i⦆<#⟦0,i+1⦆})
¬(#{⟦0,i⦆:#⟦0,i⦆<#⟦0,i+1⦆} < #{#⟦0,i⦆:#⟦0,i⦆<#⟦0,i+1⦆}+1)

Before any swap,
Bob is in room 0
>
After swaps 0⇄1 1⇄2 ... k-1⇄k and
before swap k⇄k+1
Bob is in room k
>
⎛ If Bob is in room k
⎜ them k⇄k+1 is not swapped
⎜
⎜ If Bob is in any room
⎜ then not all swaps are swapped

>
Bob is placed within
the dark parts of the matrix.

Then not all swaps are swapped

⎜ For any claims P and Q
⎜ P⇒¬Q and Q⇒¬P are equally.true and equally.false
⎜
⎜ If all swaps are swapped
⎝ them Bob is not in any room.

>
Bob comes to rest within the dark part.

Then not all swaps are swapped

Exchange of two entities does never result in
only one of them.

>
Two is finite.
>
How.many there are of
all swaps, all rooms, or all finite.ordinals
is more than any finite.ordinal,
is more.than.finite.

>
Every exchange happens between two sites.

Yes.
Always between two visibleᵂᴹ sites.
And no single swap disappears Bob.
However,
these aren't single swaps.
Infinitely.many is different from finitely.many.
For each finitely.many, there are more.

After all swaps,
⎛ Bob has never swapped into any darkᵂᴹ room,
⎝ Bob has swapped out of any visibleᵂᴹ room,
Bob is nowhere.

>
But Bob is never out of the matrix
because
there is no exchange at all between outside and inside.

Yes, Bob never exits the matrix.
However, after all swaps, Bob isn't in the matrix.
Therefore, infinite is different.