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

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Sujet : Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.math
Date : 13. Mar 2025, 20:41:12
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <8e684425-63cb-4d9c-9124-ee07acdd9017@att.net>
References : 1 2 3 4 5 6 7 8 9
User-Agent : Mozilla Thunderbird
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 secret is that
your infiniteᵂᴹ and our infiniteⁿᵒᵗᐧᵂᴹ
mean different things.
Your "No" responds to infiniteᵂᴹ,
but I wrote infiniteⁿᵒᵗᐧᵂᴹ.

Infinite is not _only_ bigger. It's different.
>
It obeys logic or it is of no value.
Consider the value of infiniteⁿᵒᵗᐧᵂᴹ
Some preliminaries to clarify finiteⁿᵒᵗᐧᵂᴹ
It's good to have such things laid out clearly.
I don't expect you (WM) to object to these points,
but who knows.
⎛⎛ In my opinion, |A| would be
⎜⎜ harder to read than #A in
⎜⎜ this next little bit.
⎜⎝ I write #A = |A| = size of set A

⎜ For at least some sets,
⎜  fuller.by.one sets are larger.
⎜ #{1} < #{1,Bob}
⎜ #{1,2} < #{1,2,Bob}
⎜ #{unicorns} < #{unicorns.and.Bob}

⎜⎛ Cᣕᶜ is fuller.by.one than C
⎜⎜ Cᣕᶜ = C∪{c} ≠ C
⎜⎜ {1,Bob},{1,2,Bob},{unicorns.and.Bob}
⎜⎜ are examples of {1}ᣕᶜ,{1,2}ᣕᶜ,{unicorns}ᣕᶜ
⎜⎜
⎜⎜ Sets for which
⎜⎜ fuller.by.one sets are larger
⎜⎝ are finiteⁿᵒᵗᐧᵂᴹ.

⎜ Consider the set of set.sizes such that
⎜  fuller.by.one sets are larger.
⎜ Consider the set {#C:#C<#Cᣕᶜ}
⎜ {#C:#C<#Cᣕᶜ} is the set of finiteⁿᵒᵗᐧᵂᴹ set.sizes.

⎜ If  #A < #Aᣕᵃ
⎜ (A has fuller.by.one Aᣕᵃ which are larger)
⎜ then set.size #A is in {#C:#C<#Cᣕᶜ}
⎜ and the other way 'round, too.
⎜  #A < #Aᣕᵃ  ⇔  #A ∈ {#C:#C<#Cᣕᶜ}

⎜ For #A < #Aᣕᵃ
⎜ the size of the set of smaller set sizes = #A
⎜  #{#C:#C<#A<#Aᣕᵃ} = #A
⎜⎛ For example,
⎜⎝ #{Bob,Kevin} = 2 = #{0,1}

⎜ No subset is larger than its superset.
⎜ A ⊆ B  ⇒  #A ≤ #B

⎜ For each #A ∈ {#C:#C<#Cᣕᶜ}
⎝  #A = #{#C:#C<#A<#Aᣕᵃ} ≤ #{#C:#C<#Cᣕᶜ})
Consider the value of infiniteⁿᵒᵗᐧᵂᴹ
⎛ A is smaller than B  iff
⎜ fuller.by.one Aᣕᵃ is smaller than fuller.by.one Bᣕᵇ
⎜ #A < #B  ⇔  #Aᣕᵃ < #Bᣕᵇ

⎜ Let B = Aᣕᵃ
⎜ #A < #Aᣕᵃ  ⇔  #Aᣕᵃ < #Aᣕᵃᵇ
⎜ A is finiteⁿᵒᵗᐧᵂᴹ  iff  larger Aᣕᵃ is finiteⁿᵒᵗᐧᵂᴹ
⎝ There is no largest finiteⁿᵒᵗᐧᵂᴹ set.
⎛ #A ∈ {#C:#C<#Cᣕᶜ}  ⇒
⎜ #A < #Aᣕᵃ  ⇒
⎜ #Aᣕᵃ < #Aᣕᵃᵇ  ⇒
⎜ #Aᣕᵃ ∈ {#C:#C<#Cᣕᶜ}

⎜ The set of finiteⁿᵒᵗᐧᵂᴹ sizes is inductive.
⎜ #A ∈ {#C:#C<#Cᣕᶜ}  ⇒  #Aᣕᵃ ∈ {#C:#C<#Cᣕᶜ}
⎝ #{} ∈ {#C:#C<#Cᣕᶜ}
⎛ For each #A ∈ {#C:#C<#Cᣕᶜ}
⎜⎛ #A ≤ #{#C:#C<#Cᣕᶜ})
⎜⎜ #Aᣕᵃ ≤ #{#C:#C<#Cᣕᶜ})
⎜⎜ #A < #Aᣕᵃ ≤ #{#C:#C<#Cᣕᶜ}
⎜⎝ #A ≠ #{#C:#C<#Cᣕᶜ}

⎜ #{#C:#C<#Cᣕᶜ} ∉ {#C:#C<#Cᣕᶜ}
⎜ The set of finiteⁿᵒᵗᐧᵂᴹ set.sizes does not have
⎜ a finiteⁿᵒᵗᐧᵂᴹ set size.

⎜⎛ It seems as though that contradicts
⎜⎜ your (WM's) idea of a proof.by.induction.
⎜⎜
⎜⎜ It does not contradict
⎜⎜ a proof that a subset is the whole.set
⎜⎜ by proving that that subset is inductive,
⎜⎜ for a whole.set which is known to be
⎜⎜ its.own.only.inductive.subset,
⎜⎜ which is what's more commonly called
⎝⎝ a proof.by.induction.
⎛⎛ #{#C:#C<#Cᣕᶜ} ∉ {#C:#C<#Cᣕᶜ}
⎜⎜
⎜⎝ ¬(#{#C:#C<#Cᣕᶜ} < #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ)

⎜⎛ {#C:#C<#Cᣕᶜ} ⊆ {#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ
⎜⎜
⎜⎝ ¬(#{#C:#C<#Cᣕᶜ} > #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ)

⎝ #{#C:#C<#Cᣕᶜ} = #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ
Consider the value of infiniteⁿᵒᵗᐧᵂᴹ
{#C:#C<#Cᣕᶜ} is an infiniteⁿᵒᵗᐧᵂᴹ set.
As an infiniteⁿᵒᵗᐧᵂᴹ set,
{#C:#C<#Cᣕᶜ} is the same size as
fuller.by.one and emptier.by.one sets,
which is why
enough size.conserving swaps
can erase Bob from {#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ:
size is conserved
going from {#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ to {#C:#C<#Cᣕᶜ}

Date Sujet#  Auteur
12 Mar 25 * The existence of dark numbers proven by the thinned out harmonic series451WM
12 Mar 25 `* Re: The existence of dark numbers proven by the thinned out harmonic series450Alan Mackenzie
12 Mar 25  `* Re: The existence of dark numbers proven by the thinned out harmonic series449WM
12 Mar 25   `* The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]448Alan Mackenzie
12 Mar 25    +* Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]444WM
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19 Mar 25    ii  iii  i i i i i i              `* Re: The non-existence of "dark numbers"253WM
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20 Mar 25    ii  iii  i i i i i i                `* Re: The non-existence of "dark numbers"251WM
20 Mar 25    ii  iii  i i i i i i                 `* Re: The non-existence of "dark numbers"250Jim Burns
20 Mar 25    ii  iii  i i i i i i                  `* Re: The non-existence of "dark numbers"249WM
20 Mar 25    ii  iii  i i i i i i                   `* Re: The non-existence of "dark numbers"248Jim Burns
21 Mar 25    ii  iii  i i i i i i                    `* Re: The non-existence of "dark numbers"247WM
21 Mar 25    ii  iii  i i i i i i                     `* Re: The non-existence of "dark numbers"246Jim Burns
21 Mar 25    ii  iii  i i i i i i                      `* Re: The non-existence of "dark numbers"245WM
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12 Mar 25    +* Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]2FromTheRafters
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