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

On 1/5/2025 6:07 AM, WM wrote:

On 04.01.2025 17:20, Jim Burns wrote:

On 1/4/2025 3:42 AM, WM wrote:

On 1/3/2025 3:56 PM, Jim Burns wrote:

All finite.ordinals removed from
the set of each and only finite.ordinals
leaves the empty set.

>
But removing
every ordinal that you can define
(and all its predecessors) from ℕ leaves
almost all ordinals in ℕ.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

>
ℕ is the set of each and only finite.ordinals.

>
Yes.

Anything which is a finite.ordinal,
darkᵂᴹ and visibleᵂᴹ, is in ℕ
Nothing which is not a finite.ordinal,
darkᵂᴹ or visibleᵂᴹ, is in ℕ
Q. What is a finite.ordinal?
⎛ It is an element of ℕ
⎜
⎜ It is an ordinal,
⎜ one of the (well.ordered) ordinals.
⎜
⎜ It is
⎜ larger.by.one than emptier.by.one
⎜  ( #⟦0,k⦆ > #(⟦0,k⦆\⦃0⦄)
⎜ or it is emptiest.
⎝  ( k = 0 = ⦃⦄

|ℕ| := ℵ₀ = |ℕ\{0}| = |ℕ\{0,1}| = ... =
|ℕ\{0,1,...,n}| = ...
>
The sequence of end.segments of ℕ
grows emptier.one.by.one but
it doesn't grow smaller.one.by.one.

>
It does
but you cannot give the numbers
because they are dark.

⎛ Among the ordinals,
⎜ for each set A of them,
⎜ A holds its first, A[1]
⎜  (excepting A=⦃⦄ which holds none)
⎜
⎜ Among the finite.ordinals
⎜  (which are ordinals),
⎜ for each set B of them,
⎜ its first B[1] has B[1]-1 not.in.B next.before.B[1]
⎜  (excepting B[1]=0 which has none before it)
⎜
⎜ For set C of the finite.ordinals,
⎜ if
⎜ (C[1] in C and C[1]-1 not.in C) is contradictory
⎜  (which is to say, impossible)
⎜ and C[1]≠0
⎝ then C=⦃⦄
We know that
all of that is true.without.exception
among the finite.ordinals,
whether.or.not I can giveᵂᴹ the finite.ordinal,
whether.or.not it's darkᵂᴹ.

The sequence of end.segments of ℕ
grows emptier.one.by.one but
it doesn't grow smaller.one.by.one.

>
It does
but you cannot give the numbers
because they are dark.

Not.giving numbers doesn't prevent us from
making claims which we know are without.exception.

A precise measure must detect
the loss of one element.
ℵo is no precise measure but only
another expression for infinitely many.

An accurate measure must recognize that
a set larger.than any.finite.set
is not any.finite.set.
https://en.wikipedia.org/wiki/Accuracy_and_precision
For each finite.set, there is
a finite.ordinal of the same size.
For each finite.ordinal, there is
a larger.by.one finite.ordinal,
and
it and its priors are a subset of
the set of all finite.ordinals.
For each finite.set, there is
a larger.than.that subset of
the set of all finite.ordinals.
For each finite.set, that set is not
the set of all finite.ordinals.
The set of all finite.ordinals is not
any.finite.set.
Q. What is a finite set?
A finite set
is smaller.by.one than emptier.by.one sets
  (excepting {} for which no emptier sets exist).
The set of finite.ordinals is not
any finite.set.
The set of finite.ordinals is not
smaller.by.one than emptier.by.one sets...
...which is
accurate (and weird) but impreciseᵂᴹ
versus
your actualᵂᴹ infinity, which is
preciseᵂᴹ (and not.weird(?)) but inaccurate.