Re: The non-existence of "dark numbers"

On 3/19/2025 3:19 PM, WM wrote:

On 19.03.2025 16:18, Jim Burns wrote:

On 3/18/2025 12:58 PM, WM wrote:

On 18.03.2025 17:34, Jim Burns wrote:

After all the swaps,
without ever disappearing into
anywhere other than a finite set,
Bob disappears out of all finite sets.

>
No, that is impossible.

>
Then there are subsets larger than their sets.

>
No, there are dark numbers. That is enough.

⎛ A set is WM.sized ==
⎜ membership.change.by.one changes size.
⎜ #A < #Aᣕᵃ
⎜
⎜ {#C:WM.sized.C} is the set of all WM.sizes.
⎜
⎜ If, for set A,
⎜ membership.change.by.one changes size,
⎜ then, for set Aᣕᵃ,
⎜ membership.change.by.one changes size.
⎝ #A e {#C:WM.sized.C}  ⇔  #Aᣕᵃ e {#C:WM.sized.C}
Does membership.change.by.one
change the size of {#C:WM.sized.C} ?
⎛ If size changes, then,
⎜ for size #{#C:WM.sized.C} = 𝔊
⎜ 𝔊,𝔊+1 ∈ {#C:WM.sized.C},
⎜ {#C:#C<𝔊+1} ⊆ {#C:WM.sized.C}
⎝ {#C:#C<𝔊+1} > {#C:WM.sized.C}
⎛ If size #{#C:WM.sized.C} doesn't change, then
⎜ the swaps which disappear Bob
⎝ don't change the size of the set.