Re: Replacement of Cardinality

On 8/30/2024 8:53 AM, WM wrote:

Le 29/08/2024 à 19:56, Jim Burns a écrit :

On 8/29/2024 9:38 AM, WM wrote:

That treats the potentially infinite collection.

>
Call it ℕᴾᴵꟲ.
The name doesn't matter.
>
There is no natural number not.in ℕᴾᴵꟲ.

⎛ Assume otherwise.
⎜ Assume darkᴹᵂ 𝔊 ≥ᵉᵃᶜʰ ℕᴾᴵꟲ
⎜
⎜ ⅟ℕᴾᴵꟲ ᵉᵃᶜʰ≥ ⅟𝔊 > 0
⎜
⎜ β = greatest.lower.bound.⅟ℕᴾᴵꟲ
⎜ β > α  ⇒  ⅟ℕᴾᴵꟲ ᵉᵃᶜʰ≥ α
⎜ γ > β  ⇒  ⅟ℕᴾᴵꟲ ᵉˣⁱˢᵗˢ≱ γ
⎜ β ≥ ⅟𝔊 > 0
⎜
⎜ 2⋅β > β > ½⋅β > 0
⎜ β > ½⋅β  ⇒  ⅟ℕᴾᴵꟲ ᵉᵃᶜʰ≥ ½⋅β
⎜ 2⋅β > β  ⇒  ⅟ℕᴾᴵꟲ ᵉˣⁱˢᵗˢ≱ 2⋅β
⎜
⎜ However,
⎜ ⅟ℕᴾᴵꟲ ᵉˣⁱˢᵗˢ≱ 2⋅β
⎜ visibleᵂᴹ ⅟k ≱ 2⋅β
⎜ visibleᵂᴹ ¼⋅⅟k ≱ ½⋅β
⎜ ⅟ℕᴾᴵꟲ ᵉˣⁱˢᵗˢ≱ ½⋅β
⎜
⎜ ...which contradicts
⎝ ⅟ℕᴾᴵꟲ ᵉᵃᶜʰ≥ ½⋅β
Therefore,
darkᴹᵂ 𝔊 ≱ᵉˣⁱˢᵗˢ ℕᴾᴵꟲ

Maybe if Bob can disappear.

⎛ ∀S ⊆ ℕ: S ≠ {} ⇔ ∃k ∈ S: k = min.S
⎜
⎜ ∀k ∈ ℕ: k ≠ 0 ⇔ ∃j ∈ ℕ: j+1 = k
⎜
⎝ ∀j ∈ ℕ: ∃k ∈ ℕ: j+1 = k
Move Bob into room 0.
In order, swap guests in rooms n and n+1.
If Bob is in room 𝔊, not all swaps are swapped.
If all swaps are swapped, Bob is not in room 𝔊.
Or in any room.
'Bye, Bob.

But logic prevents that.

The next time you go further than saying
"There is logic" to saying
"Here is the logic..."
will be the first time.