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

On 12/22/2024 5:24 PM, WM wrote:

On 22.12.2024 22:22, Jim Burns wrote:

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

A set is finiteᵂᴹ if it contains
a visibleᵂᴹ naturalᵂᴹ number of elements.
All naturalᵂᴹ numbers are finiteᵂᴹ but
the realm of darkᵂᴹ numbers
appears like infinityᵂᴹ.

 Set A is finiteⁿᵒᵗᐧᵂᴹ if A∪{x} is larger, for A∌x
Set Y is infiniteⁿᵒᵗᐧᵂᴹ if Y∪{x} isn't larger, for Y∌x
 ℕⁿᵒᵗᐧᵂᴹ is the set of finiteⁿᵒᵗᐧᵂᴹ.cardinals.
 which means
ℕⁿᵒᵗᐧᵂᴹ is the set of cardinals #A of
 sets A smaller.than A∪{x} for A∌x
 For each finiteⁿᵒᵗᐧᵂᴹ set A,
its cardinality #A is in ℕⁿᵒᵗᐧᵂᴹ
 For each set Y without #Y in ℕⁿᵒᵗᐧᵂᴹ
Y is infiniteⁿᵒᵗᐧᵂᴹ, and
Y∪{x} isn't larger than Y, for Y∌x

The function E(n) decreases
from infinity to zero
by single steps of height 1
like the function NUF(x) increases
by single steps of height 1.
Set theorists must accept
magic steps of infiniteᵂᴹ size or
refuse to describe these transitions at all.

>
Alternatively,
set theorists could continue to talk about
what set theorists talk about, instead of
what you (WM) talk about.

>
They do not wish to recognize that
their theory is self-contradictory. But perhaps students would be interested.
I tell them the following story.
>
The function Eᵂᴹ(n) decreases
from infinityᵂᴹ to zero
because in set theory ℕᵂᴹ \ {1, 2, 3, ...} =  { }
is an accepted formula.

ℕᵂᴹ is larger.than emptier.by.one sets,
and so, too, is smaller.than fuller.by.one sets.
and so, #ℕᵂᴹ is in ℕⁿᵒᵗᐧᵂᴹ
ℕᵂᴹ is finiteⁿᵒᵗᐧᵂᴹ
#ℕⁿᵒᵗᐧᵂᴹ >ᵉᵃᶜʰ ℕⁿᵒᵗᐧᵂᴹ
#ℕⁿᵒᵗᐧᵂᴹ ≠ᵉᵃᶜʰ ℕⁿᵒᵗᐧᵂᴹ
#ℕⁿᵒᵗᐧᵂᴹ is not.in ℕⁿᵒᵗᐧᵂᴹ
ℕⁿᵒᵗᐧᵂᴹ is not.larger.than emptier.by.one sets.
ℕⁿᵒᵗᐧᵂᴹ is infiniteⁿᵒᵗᐧᵂᴹ
ℕⁿᵒᵗᐧᵂᴹ is a proper superset of ℕᵂᴹ
∀n ∈ ℕⁿᵒᵗᐧᵂᴹ:
Eⁿᵒᵗᐧᵂᴹ(n+1) = Eⁿᵒᵗᐧᵂᴹ(n)\{n+1}
There are sets emptier.by.one than Eⁿᵒᵗᐧᵂᴹ(n)
∀n ∈ ℕⁿᵒᵗᐧᵂᴹ:
#Eⁿᵒᵗᐧᵂᴹ(n+1) >ᵉᵃᶜʰ ℕⁿᵒᵗᐧᵂᴹ
#Eⁿᵒᵗᐧᵂᴹ(n+1) ≠ᵉᵃᶜʰ ℕⁿᵒᵗᐧᵂᴹ
No sets are smaller.by.one than Eⁿᵒᵗᐧᵂᴹ(n)
∀n ∈ ℕⁿᵒᵗᐧᵂᴹ:
n ∉ Eⁿᵒᵗᐧᵂᴹ(n) ⊆ ℕⁿᵒᵗᐧᵂᴹ
Finiteⁿᵒᵗᐧᵂᴹ.cardinal n is not common to
all end.segments of the finiteⁿᵒᵗᐧᵂᴹ.cardinals.

The set ℕᵂᴹ can get empty
by subtracting its elements.

#ℕᵂᴹ > #(ℕᵂᴹ\{0})
#ℕᵂᴹ ∈ ℕⁿᵒᵗᐧᵂᴹ
ℕᵂᴹ is finiteⁿᵒᵗᐧᵂᴹ
ℕᵂᴹ is a proper subset of ℕⁿᵒᵗᐧᵂᴹ

Either this is possible one by one,
then finite endsegments do exist,

For finiteⁿᵒᵗᐧᵂᴹ ℕᵂᴹ
all its end.segments are finiteⁿᵒᵗᐧᵂᴹ
For infiniteⁿᵒᵗᐧᵂᴹ ℕⁿᵒᵗᐧᵂᴹ
each finiteⁿᵒᵗᐧᵂᴹ.cardinal
is individual, can be removed.
Infinitelyⁿᵒᵗᐧᵂᴹ.many of them cannot be removed
finitelyⁿᵒᵗᐧᵂᴹ.

or it is only possible be removing
(after the first elements one by one)
the remaining elements collectively.
This shows the existence of
numbers which can be handled collectively only.

The set ℕⁿᵒᵗᐧᵂᴹ of finiteⁿᵒᵗᐧᵂᴹ.cardinals holds
only cardinals which are finiteⁿᵒᵗᐧᵂᴹ, of which
there are more.than any.finiteⁿᵒᵗᐧᵂᴹ.cardinal.many.

How should we call them?

Infinite.finite.cardinals.
⎛ How we finiteⁿᵒᵗᐧᵂᴹ beings are capable of
⎜ learning about infinitelyⁿᵒᵗᐧᵂᴹ.many is by
⎜ curating a finiteⁿᵒᵗᐧᵂᴹ sequence of claims about
⎜ an indefinite one of those infinitelyⁿᵒᵗᐧᵂᴹ.many,
⎜ in which each claim is known by us to be
⎜ true.or.not.first.false.
⎜
⎜ We refer collectively to each individual.
⎜
⎜ If referring is handling, then Yippeee!
⎜ I am handling all the money in the world.
⎝ Grovel before me, you peasants!

Another approach is to call [prove] the empty set
the limit of the sequence E(n).
But note that a limit is a set [JB:]
⎡ in.which is each element in each set of
⎢ almost.all of infinitelyⁿᵒᵗᐧᵂᴹ.many sets
⎢ and
⎢ not.in.which is each element not.in each set of
⎣ almost.all of infinitelyⁿᵒᵗᐧᵂᴹ.many sets
between which [it] and all terms of the sequence
nothing fits.
Therefore the limit empty set causes
sets with few elements only.

The empty limit set is caused by
each finiteⁿᵒᵗᐧᵂᴹ.cardinal being not.in
almost all of the end.segments
(in only finitely.many end.segment.exceptions)