Re: The non-existence of "dark numbers"

On 3/26/2025 2:37 PM, WM wrote:

On 26.03.2025 07:24, Jim Burns wrote:

On 3/25/2025 6:12 PM, WM wrote:

On 25.03.2025 21:26, Jim Burns wrote:

On 3/25/2025 4:17 PM, WM wrote:

On 25.03.2025 18:46, Jim Burns wrote:

It's the same way in which we know that
a square has four corners

>
and that lossless exchanges are lossless.

>
and that a set larger than
any set different.sized.by.one from other sets
is not
any set different.sized.by.one from other sets.

>
Irrelevant.
Your waffle does not change basic logic.

>
A set larger than
any set for which my waffle does not change basic WM.logic
is
a set for which my waffle changes basic WM.logic.

A set Y different from
  each set A without Property I
is
a set Y with Property I.
  (∀A: ¬I(A) ⇒ A≠Y)  ⇒  I(Y)

Nothing of that kind does exist.

Consider the set {#C:#C<#Cᣕᶜ} of all finite sizes.
⎛ By 'finite', I mean
⎜ a set from which
⎜ some other sets are different.by.one in size.
⎜ Here's hoping that you agree that finite sets exist.
⎜
⎜⎛ Cᣕᶜ means a set fuller.by.one than C
⎜⎝ Cᣕᶜ = C∪{c} ≠ C
⎜
⎜ Even though you (WM) will quote Wikipedia,
⎜ you mean something else by 'finite', something like
⎜ 'the opposite of reallyreallyreallybig'.
⎜
⎜ That's why I've written {#C:#C<#Cᣕᶜ}
⎜ This is my (our) 'finite', not your 'finite'.
⎜
⎜ A set Y with Property I does not have a finite size.
⎜ Its size #Y is not in {#C:#C<#Cᣕᶜ}
⎜ I(Y)  ⇔  #Y ∉ {#C:#C<#Cᣕᶜ}
⎜
⎜ A set A without Property I has a finite size.
⎜ Its size #A is in {#C:#C<#Cᣕᶜ}
⎜ ¬I(A)  ⇔  #A ∈ {#C:#C<#Cᣕᶜ}
⎜
⎜ Membership of a size in {#C:#C<#Cᣕᶜ} is determined by #C<#Cᣕᶜ
⎜ ¬I(A)  ⇔  #A ∈ {#C:#C<#Cᣕᶜ}  ⇔  #A < #Aᣕᵃ
⎜ I(Y)  ⇔  #Y ∉ {#C:#C<#Cᣕᶜ}  ⇔  #Y = #Yᣕʸ
⎜
⎜⎛ It is never the case that
⎝⎝ (finite;infinite) #J > #Jᣕʲ
Here is where WM.logic starts to fall apart:
#J < #K  ⇔  #Jᣕʲ < #Kᣕᵏ
(finite;infinite) J is smaller than (finite;infinite) K
iff
fuller.by.one Jᣕʲ is smaller than fuller.by.one Kᣕᵏ
⎛ In the interest of conciseness,
⎝ I will wait for a request before showing the proof.
#J < #K  ⇔  #Jᣕʲ < #Kᣕᵏ
Let K = Jᣕʲ
#J < #Jᣕʲ  ⇔  #Jᣕʲ < #Jᣕʲᵏ
Also,
#J = #Jᣕʲ  ⇔  #Jᣕʲ = #Jᣕʲᵏ
If J is finite (#J < #Jᣕʲ)
then Jᣕʲ is finite (#Jᣕʲ < #Jᣕʲᵏ)
If Jᣕʲ is infinite (#Jᣕʲ = #Jᣕʲᵏ)
then J is infinite (#J = #Jᣕʲ)
There is no single.step up from finite to infinite.
There is no single.step down from infinite to finite.
WM.logic still needs to accept that.

Nothing of that kind does exist.

Consider the set {#C:#C<#Cᣕᶜ} of all finite sizes.
For each finite size #A, there is
a subset {#C:#C<#A} of {#C:#C<#Cᣕᶜ} of size #A
For each finite size #A, there is
a larger finite size #Aᣕᵃ and
a subset of {#C:#C<#Cᣕᶜ} larger than #A
For each finite size #A,
#A cannot be the size of {#C:#C<#Cᣕᶜ}
because
{#C:#C<#Cᣕᶜ} can't have a subset larger than it.
#{#C:#C<#Cᣕᶜ} ∉ {#C:#C<#Cᣕᶜ}
#{#C:#C<#Cᣕᶜ} = #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ

"WM-logic" says that
lossless exchanges at finite steps
are lossless

For a set larger than
any set such that
  emptier.by.one and smaller.by.one are the same,
emptier.by.one and smaller.by.one aren't the same.

and is the foundation of rational thinking.

"Rational thinking", "logic", "mathematics",
"geometry", "not matheology", ...
It's much easier to write advertising copy
than to find an actual argument.