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

On 1/14/2025 4:07 AM, WM wrote:

On 13.01.2025 20:31, Jim Burns wrote:

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

[...]

>
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

>
All that is true in potential infinity,
however it is wrong in completed infinity.

Your infinity.completingᵂᴹ darkᵂᴹ numbers 𝔻
have negative cardinality.
|𝔻| = #𝔻 = -∞
Consider the finiteˢᵉᵗ sets.
⎛ A non.empty finiteˢᵉᵗ linearly.ordered is
⎜ two.ended and each of its non.empty subsets is
⎜ two.ended.
⎜
⎜ Sets emptier.by.one and sets fuller.by.one
⎜ than a finiteˢᵉᵗ are sized differently from it.
⎜ They are finitesˢᵉᵗ themselves.
⎜
⎜ For each finiteˢᵉᵗ set, there exists
⎝ a larger finiteᵒʳᵈ ordinal.
⎛ A set larger.than each finiteˢᵉᵗ
⎜ is not itself any of the finitesˢᵉᵗ.
⎜
⎜ Claims which are true of a finiteˢᵉᵗ
⎜ because it's finiteˢᵉᵗ
⎜ are false of a not.any.finiteˢᵉᵗ set.
⎜
⎜ For example,
⎜ sets emptier.by.one and sets fuller.by.one
⎜ than a not.any.finiteˢᵉᵗ
⎜ are NOT sized differently from it.
⎜
⎜ A not.any.finiteˢᵉᵗ is an infiniteˢᵉᵗ.
⎜ As you (WM) explain,
⎝ it is a potentiallyᵂᴹ infiniteˢᵉᵗ.
⎛ ℕ is the least.upper.bound of finitesᵒʳᵈ.
⎜
⎜ For each finiteˢᵉᵗ,
⎜ a larger finiteᵒʳᵈ exists,
⎜ and the larger LUB ℕ exists, too.
⎜
⎜ ℕ is not.any.finiteˢᵉᵗ.
⎜ ℕ is infiniteˢᵉᵗ.
⎜ As you (WM) explain,
⎝ ℕ is potentiallyᵂᴹ infiniteˢᵉᵗ.
⎛ An actuallyᵂᴹ infiniteˢᵉᵗ 𝔸 is
⎜ a potentiallyᵂᴹ infiniteˢᵉᵗ ℙ which has been
⎜ completedᵂᴹ by an appropriate darkᵂᴹ 𝔻
⎜ such that 𝔸 isn't potentiallyᵂᴹ infiniteˢᵉᵗ.
⎜
⎜ Potentialᵂᴹ ℙ is larger.than.any.finiteˢᵉᵗ.
⎜ Actualᵂᴹ 𝔸 is not larger.than.any.finiteˢᵉᵗ.
⎝ 𝔸 = ℙ∪𝔻
⎛ Potentialᵂᴹ ℙ is larger.than.any.finiteˢᵉᵗ.
⎜ Actualᵂᴹ 𝔸 is not larger.than.any.finiteˢᵉᵗ.
⎜ 𝔸 = ℙ∪𝔻,  #(ℙ∩𝔻) = 0
⎜
⎜ There is a finiteˢᵉᵗ F: #𝔸 ≤ #F
⎜ ℙ is larger than finiteˢᵉᵗ F
⎜
⎜ #𝔸 ≤ #F < #ℙ
⎜
⎜ #𝔸 = #ℙ + #𝔻 - #(ℙ∩𝔻)
⎜
⎝ #ℙ + #𝔻 < #P
𝔻 has negative cardinality.
----

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

>
No.
They fit below ω

>
In completed infinity
all available places are occupied.

In each of our sets,
each element has an available space, and
only its elements have available spaces.
A place in a set is occupied by virtue of
its element being in the set.
In each of our sets,
each of its elements is in the set,
each available place is occupied.
A potentiallyᵂᴹ infiniteˢᵉᵗ set,
the same as any other set,
has all available places occupied
and is completeᵂᴹ.

Half are new.

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