Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]

On 3/14/2025 10:33 AM, WM wrote:

On 13.03.2025 20:41, Jim Burns wrote:

On 3/13/2025 6:45 AM, WM wrote:

On 13.03.2025 01:43, Jim Burns wrote:

A single (lossless) exchange cannot delete an O
Finitely.many (lossless) exchanges cannot delete an O
>
Infinitely.many (lossless) exchanges can delete an O

>
No.

>
Your "No" responds to infiniteᵂᴹ,
but I wrote infiniteⁿᵒᵗᐧᵂᴹ.

>
My "No" responds to every finity and every infinity.

It doesn't respond to this definition:
⎛ a finiteⁿᵒᵗᐧᵂᴹ set A  has
⎜ fuller.by.one sets Aᣕᵃ which are larger.
⎜  finiteⁿᵒᵗᐧᵂᴹ A: #A < #Aᣕᵃ
⎜ and
⎜ an infiniteⁿᵒᵗᐧᵂᴹ set Y has
⎜ fuller.by.one sets Yᣕʸ which are not larger.
⎝  infiniteⁿᵒᵗᐧᵂᴹ Y: #Y = #Yᣕʸ
⎛ ¬(#A > #Aᣕᵃ)
⎝ ¬(#Y > #Yᣕʸ)
lemma:
⎛ A is smaller than B  iff
⎜ fuller.by.one Aᣕᵃ is smaller than fuller.by.one Bᣕᵇ
⎝  #A < #B  ⇔  #Aᣕᵃ < #Bᣕᵇ
If you deny the definition,
then you refuse to hear what I'm saying,
you choose to hear infiniteᵂᴹ not infiniteⁿᵒᵗᐧᵂᴹ
If you deny the lemma
  #A < #B  ⇔  #Aᣕᵃ < #Bᣕᵇ
I have a proof to show you.
----
  #A < #B  ⇔  #Aᣕᵃ < #Bᣕᵇ
Let B = Aᣕᵃ
  #A < #Aᣕᵃ  ⇔  #Aᣕᵃ < #Aᣕᵃᵇ
If A is finiteⁿᵒᵗᐧᵂᴹ
then Aᣕᵃ is finiteⁿᵒᵗᐧᵂᴹ and larger.
There is no largest finiteⁿᵒᵗᐧᵂᴹ set.
By similar reasoning, for infiniteⁿᵒᵗᐧᵂᴹ Y and
emptier.by.one Yᐠʸ and emptier.by.two Yᐠʸᶻ
  #Y = #Yᐠʸ  ⇔  #Yᐠʸ = #Yᐠʸᶻ
If Y is infiniteⁿᵒᵗᐧᵂᴹ,
removing singles doesn't shrink Y.
What you must deny in order to say "No" is
  #A < #B  ⇔  #Aᣕᵃ < #Bᣕᵇ
----
{#C:#C<#Cᣕᶜ} is the set of finiteⁿᵒᵗᐧᵂᴹ set.sizes.
{#C:#C<#Cᣕᶜ} = ℕⁿᵒᵗᐧᵂᴹ
For each finite set A, #A < #Aᣕᵃ
set.size #A is in {#C:#C<#Cᣕᶜ}
No set.size in {#C:#C<#Cᣕᶜ} is #{#C:#C<#Cᣕᶜ}
Otherwise,
there would be subsets of {#C:#C<#Cᣕᶜ} which
were larger than {#C:#C<#Cᣕᶜ}
(There is no negative cardinality.)
¬(#{#C:#C<#Cᣕᶜ} < #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ)
#{#C:#C<#Cᣕᶜ} = #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ
This is why
sufficiently.many size.preserving swaps
can erase Bob:
erasing Bob preserves the size of {#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ

Logic never ceases to be valid.
Lossless exchange is lossless.

  #A < #B  ⇔  #Aᣕᵃ < #Bᣕᵇ