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.
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