Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, effectively)

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, ....