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NNTP-Posting-Date: Fri, 27 Dec 2024 22:24:50 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary, effectively)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Fri, 27 Dec 2024 14:24:13 -0800
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On 12/27/2024 01:00 PM, Jim Burns wrote:
> On 12/27/2024 5:14 AM, WM wrote:
>> On 26.12.2024 19:41, Jim Burns wrote:
>
>>> [...]
>>
>> A limit is a set S​͚  such that nothing fits
>> between it and all sets of the sequence.
>
> Sₗᵢₘ is _almost_ each set in ⟨Sₙ⟩ₙ₌᳹₀
>
> ⋂ₙ₌᳹ₖ⟨Sₙ⟩ holds each element which
> is in each set of ⟨Sₙ⟩ₙ₌᳹₀ with
>   up.to.finite.k exceptions
> ⋃ₖ₌᳹₀⋂ₙ₌᳹ₖ⟨Sₙ⟩ holds each element which
> is in each set of ⟨Sₙ⟩ₙ₌᳹₀ with
>   up.to.finitely.many exceptions
>
> ⋃ₖ₌᳹₀⋂ₙ₌᳹ₖ⟨Sₙ⟩  ⊆  Sₗᵢₘ
>
> ⋂ₙ₌᳹ₖ⟨Sₙᒼ⟩ holds each complement.element which
> is in each complement.set of ⟨Sₙᒼ⟩ₙ₌᳹₀ with
>   up.to.finite.k exceptions
> ⋃ₖ₌᳹₀⋂ₙ₌᳹ₖ⟨Sₙᒼ⟩ holds each complement.element which
> is in each complement.set of ⟨Sₙᒼ⟩ₙ₌᳹₀ with
>   up.to.finitely.many exceptions
>
> ⋃ₖ₌᳹₀⋂ₙ₌᳹ₖ⟨Sₙᒼ⟩  ⊆  Sₗᵢₘᒼ
>
> Sₗᵢₘ  ⊆  (⋃ₖ₌᳹₀⋂ₙ₌᳹ₖ⟨Sₙᒼ⟩)ᒼ  =  ⋂ₖ₌᳹₀⋃ₙ₌᳹ₖ⟨Sₙ⟩
>
> ⋃ₖ₌᳹₀⋂ₙ₌᳹ₖ⟨Sₙ⟩  ⊆  Sₗᵢₘ  ⊆  ⋂ₖ₌᳹₀⋃ₙ₌᳹ₖ⟨Sₙ⟩
>
>>> The notation a​͚  or S​͚  for aₗᵢₘ or Sₗᵢₘ
>>> is tempting, but
>>> it gives the unfortunate impression that
>>> a​͚  and S​͚  are the infinitieth entries of
>>> their respective infinite.sequences.
>>> They aren't infinitieth entries.
>>> They are defined differently.
>>
>> The last natural number is finite,
>
> To be finite.cardinal k is
> for the following to be true:
> #⟦0,k⦆ < #(⟦0,k⦆∪⦃k⦄)  ∧  #⟦0,k+1⦆ < #(⟦0,k+1⦆∪⦃k+1⦄)
>
> To be finite.cardinal k is
> to be smaller.than finite.cardinal k+1
>
> To be finite.cardinal k is
> to not.be the largest finite.cardinal.
>
>> But like all dark numbers
>> it has no FISON
>
> To be a finite.cardinal is
> to have a finite set of prior cardinals, ie,
> to have a FISON.
>
> Therefore,
> to be a finite.cardinal and darkᵂᴹ is
> to be self.contradictory, and to not.exist.
>
>>>>> #E(n+2) isn't any of the finite.cardinals in ℕ
>>>>
>>>> It is an infinite number but
>>>> even infinite numbers differ like |ℕ| =/= |ℕ| + 1.
>>>
>>> Infiniteᵂᴹ numbers which differ like |ℕ| ≠ |ℕ| + 1.
>>> are finiteⁿᵒᵗᐧᵂᴹ numbers.
>>
>> No.
>> They are invariable numbers like ω and ω+1.
>
> ω is
> the set of (well.ordered) ordinals k such that
> #⟦0,k⦆ ≠ #(⟦0,k⦆∪⦃k⦄)
> (such that k is finite)
>
> There is no k ∈ ω
>   ω = ⦃i: #⟦0,i⦆≠#(⟦0,i⦆∪⦃i⦄) ⦄
> such that
> #ω = k
>
> ¬(#ω ∈ ω)
> ¬(#⟦0,ω⦆ ≠ #(⟦0,ω⦆∪⦃ω⦄))
> #⟦0,ω⦆ = #(⟦0,ω⦆∪⦃ω⦄)
> (|ℕ| = |ℕ|+1)
>
> A separate fact is that
> ⟦0,ω⦆ ≠ ⟦0,ω⦆∪⦃ω⦄
>
>

In telecommunications, sometimes when there's
more than a 10:1 source/channel ratio, it's
said there are "effectively infinite sources",
then the opposite of that is called "limited".


Being effectively infinite then 1/x = 0,
and simplifies many formulas.


The, "almost all", or, "almost everywhere",
does _not_ equate to "all" or "everywhere",
and these days in sub-fields of mathematics
like to do with topology and the ultrafilter,
it's a usual conceit to in at least one
sense, not being "actually" correct.


Horse-shoes and hand-grenades, ....