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

On 15.01.2025 16:16, Jim Burns wrote:

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

Therefore new numbers are not accepted.

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

Potential infinity is growing. "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]

 

Half are new.

 A step is never from finite to infinite.

∀n ∈ ℕ: 2n =/= n.
Regards, WM