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

On 16.01.2025 02:23, Jim Burns wrote:

A potentiallyᵂᴹ infiniteˢᵉᵗ set
has all available places occupied,

But it can grow. Multiplying all its elements by 2 creates new elements.

the same as any other set,
which is to say,
it is (has been, will be) completeᵂᴹ.

Then new numbers must be outside.

 

"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]

For the set ℕ of all finite ordinals,
a finite ordinal larger than ℕ
doesn't exist.

Therefore multiplying all finite ordinals creates infinite ordinals.
Regards, WM