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

On 3/15/2025 4:36 AM, WM wrote:

On 15.03.2025 00:17, Jim Burns wrote:

On 3/14/2025 5:55 PM, WM wrote:

On 14.03.2025 18:29, Jim Burns wrote:

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:

>
Then it is not worthwhile to read that definition.

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

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

There is a very old anecdote told in which
Galileo Galilei tries to show the moons of Jupiter
to a representative of the pope.
The representative refuses to look at
anything his religion says can't exist.
----
I give you (WM)
'ovine' (usually 'sheep.related') and 'nonovine'
as replacements for
'finite'ⁿᵒᵗᐧᵂᴹ and 'infinite'ⁿᵒᵗᐧᵂᴹ
Set Aᣕᵃ is fuller.by.one than set A
Aᣕᵃ = A∪{a} ≠ A
Set Wᐠʷ is emptier.by.one than set W
Wᐠʷ = W\{w} ≠ W ≠ {}
Set A is ovine iff Aᣕᵃ is larger.
Set W is nonovine iff Wᐠʷ is not smaller.
For all sets, either ovine or nonovine,
#A < #B  ⇔  #Aᣕᵃ < #Bᣕᵇ
#W = #Y  ⇔  #Wᐠʷ = #Yᐠʸ
Let B = Aᣕᵃ. Let Y = Wᐠʷ.
#A < #Aᣕᵃ  ⇔  #Aᣕᵃ < #Aᣕᵃᵇ
#W = #Wᐠʷ  ⇔  #Wᐠʷ = #Wᐠʷʸ
For ovine A, fuller.by.one Aᣕᵃ is also ovine.
For nonovine W, emptier.by.one Wᐠʷ is also nonovine.
Consider a one.by.one decreasing set.sequence
with emptier.by.one successors
and fuller.by.one predecessors.
Sᵢ, Sᵢ₊₁, ..., Sₖ₋₁, Sₖ
Sᵢ₊₁ = Sᵢᐠⁱ, ...
Sₖ₋₁ = Sₖᣕᵏ, ...
If Sₖ is ovine (as, for example, {} is)
then
#Sᵢ > #Sᵢ₊₁ > ... > #Sₖ₋₁ > #Sₖ
and
Sᵢ is ovine.
If Sᵢ is nonovine
then
#Sᵢ = #Sᵢ₊₁ = ... = #Sₖ₋₁ = #Sₖ
and
Sₖ is nonovine and #Sᵢ = #Sₖ
Sₖ isn't {}
Consider the set {#C:#C<#Cᣕᶜ} of ovine set.sizes.
#A ∈ {#C:#C<#Cᣕᶜ}  ⇔  #A < #Aᣕᵃ
For each ovine set.size k ∈ {#C:#C<#Cᣕᶜ}
there is {#C:#C<k}:  #{#C:#C<k} = k
and {#C:#C<k}ᣕᵏ: #{#C:#C<k}ᣕᵏ > k
For each ovine set.size k ∈ {#C:#C<#Cᣕᶜ}
k < #{#C:#C<k}ᣕᵏ ≤ #{#C:#C<#Cᣕᶜ}
k ≠ #{#C:#C<#Cᣕᶜ}
{#C:#C<#Cᣕᶜ} has a nonovine set.size.
Consider a one.by.one decreasing set.sequence
with emptier.by.one successors
and fuller.by.one predecessors.
Sᵢ, Sᵢ₊₁, ..., Sₖ₋₁, Sₖ
Sᵢ₊₁ = Sᵢᐠⁱ, ...
Sₖ₋₁ = Sₖᣕᵏ, ...
with
Sᵢ = {#C:#C<#Cᣕᶜ}
#Sₖ = #{#C:#C<#Cᣕᶜ}
∀k ∈ {#C:#C<#Cᣕᶜ}:
|{#C:#C<#Cᣕᶜ}\{0,1,...,k}| = ℵ₀