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

On 12/22/2024 6:32 AM, WM wrote:

On 22.12.2024 11:16, Jim Burns wrote:

What does 'finite' mean?

>
A set is finite if it contains
a visible natural number of elements.
All natural numbers are finite but
the realm of dark numbers
appears like infinity.

I (JB) think that it may be that
'almost.all' '(∀)' refers concisely to
the differences in definition at
the center of our discussion.
For each finite.cardinal,
almost.all finite.cardinals are larger.
∀j ∈ ℕⁿᵒᵗᐧᵂᴹ: (∀)k ∈ ℕⁿᵒᵗᐧᵂᴹ: j < k
I think that you (WM) would deny that.
You would say, instead,
ᵂᴹ⎛ for each definable finite.cardinal
ᵂᴹ⎜ almost.all finite cardinals are larger.
ᵂᴹ⎜ ∀j ∈ ℕᵂᴹ_def: (∀)k ∈ ℕᵂᴹ: j < k
ᵂᴹ⎜ because
ᵂᴹ⎜ almost.all of ℕᵂᴹ is not.in ℕᵂᴹ_def
ᵂᴹ⎝ (∀)j ∈ ℕᵂᴹ: j ∉ ℕᵂᴹ_def
That isn't what we mean by 'finite.cardinal'.
----
For sequence ⟨aₙ⟩ₙ᳹₌₀ of points in
space 𝔸 which has open sets Sₒ ∈ 𝒪[𝔸],
aₗᵢₘ is a limit.point of ⟨aₙ⟩ₙ᳹₌₀
if
each neighborhood Sₒ ∋ aₗᵢₘ:  Sₒ ∈ 𝒪[𝔸]
holds almost.all of ⟨aₙ⟩ₙ᳹₌₀
⟨aₙ⟩ₙ᳹₌₀ ⟶ aₗᵢₘ  ⇐
⎛ ∀Sₒ ∈ 𝒪[𝔸]: Sₒ ∋ aₗᵢₘ  ⇒
⎝ (∀)aₖ ∈ ⟨aₙ⟩ₙ᳹₌₀: Sₒ ∋ a​ₖ
⎛ If 𝔸 holds no limit.points aₗᵢₘ of ⟨aₙ⟩ₙ᳹₌₀
⎜ then ⟨aₙ⟩ₙ᳹₌₀ doesn't converge in 𝔸
⎜
⎜ If, in space 𝔸 having 𝒪[𝔸],
⎜ distinct points have disjoint neighborhoods
⎜ (Hausdorff)
⎜ then
⎝ ⟨aₙ⟩ₙ᳹₌₀ has no more than one limit.point in 𝔸.
----
For sequence ⟨Sₙ⟩ₙ᳹₌₀ of sets
Sₗᵢₘ is a limit.set of ⟨Sₙ⟩ₙ᳹₌₀
if
each x ∈ Sₗᵢₘ is ∈ almost.all Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀  and
each y ∉ Sₗᵢₘ is ∉ almost.all Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀
⟨Sₙ⟩ₙ᳹₌₀ ⟶ Sₗᵢₘ  ⇐
⎛ x ∈ Sₗᵢₘ  ⇐  (∀)Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀:  x ∈ Sₖ
⎝ y ∉ Sₗᵢₘ  ⇐  (∀)Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀:  y ∉ Sₖ
---
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.

E(n+2) is
the set of all finite.cardinals > n+2
E(n+1)  =  E(n+2)∪{n+2}
E(n+2)∪{n+2} isn't larger.than E(n+2)

>
Wrong.

Almost all finite.cardinals are larger than
finite.cardinal n+1
{n+2} isn't large enough to change that.
Almost all finite.cardinals are larger than
finite cardinal n+2

#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.
The concept of 'limit' is a cornerstone of
calculus and analysis and topology.
That cornerstone rests upon 'almost.all'.
Each finite.cardinal has infinitely.more
finite.cardinals after it than before it.
If one re.defines things away from that,
it is only an odd coincidence if, after that,
any part of calculus or analysis or topology
continues to make sense.