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

On 1/15/2025 1:17 PM, WM wrote:

On 15.01.2025 16:16, Jim Burns wrote:

On 1/14/2025 4:07 AM, WM wrote:

On 13.01.2025 20:31, Jim Burns wrote:

On 1/13/2025 12:17 PM, WM wrote:

therefore creates even numbers.
They do not fit below ω.

>
No.
They fit below ω

>
In completed infinity
all available places are occupied.

In each of our sets,
each of its elements is in the set,
each available place is occupied.

>
Therefore new numbers are not accepted.

And all even numbers fit below ω
None are created.

A potentiallyᵂᴹ infiniteˢᵉᵗ set,
the same as any other set,
has all available places occupied
and is completeᵂᴹ.

>
Potential infinity is growing.

In each of our sets,
each element has an available space, and
only its elements have available spaces.
A place in a set is occupied by virtue of
its element being in the set.
In each of our sets,
each of its elements is in the set,
each available place is occupied.
A potentiallyᵂᴹ infiniteˢᵉᵗ set
has all available places occupied,
the same as any other set,
which is to say,
it is (has been, will be) completeᵂᴹ.

"In analysis we have to deal only
with the infinitely small and
the infinitely large
as a limit-notion,
as something becoming, emerging, produced,
i.e., as we put it, with the potential infinite.
But this is not the proper infinite.
That we have for instance
when we consider
the entirety of the numbers 1, 2, 3, 4, ... itself
as a completed unit, or
the points of a line as
an entirety of things which is completely available.
That sort of infinity is named actual infinite."
[D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]

A finite set has
emptier.by.one sets which are smaller.
For each finite set,
a finite ordinal larger than that set
exists.
For the set ℕ of all finite ordinals,
a finite ordinal larger than ℕ
doesn't exist.
Therefore,
the set ℕ of all finite ordinals
isn't itself finite,  and,
unlike a finite set, ℕ doesn't have
emptier.by.one sets which are smaller.
----
Finite people are able to reason about infinity
by describing an indefinite one of infinitely.many
and then supplementing the descriptive claims
with visibly not.first.false claims.
As finite people,
we have not and _cannot_ witness
the infinitely.many described.
What we can witness, instead, are
the finitely.many finite.length claims themselves,
and witness the correctness of the description of
that which we are currently discussing,
and witness the not.first.false.ness of
the other claims.
Upon witnessing all that,
we know the claims are true.