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

On 12.12.2024 16:30, joes wrote:

Am Thu, 12 Dec 2024 15:33:20 +0100 schrieb WM:

On 12.12.2024 15:23, joes wrote:

Am Thu, 12 Dec 2024 10:12:26 +0100 schrieb WM:

>

The end of the sequence is defined by ∀k ∈ ℕ : E(k+1) = E(k) \ {k}.

The sequence is endless, has no end, is infinite.

If a bijection with ℕ is possible, the sequence can be exhausted so that
no natural numbers remains in an endsegment.

For you: no, an infinite set cannot be exhausted in finite steps.
 

There are infinitely many steps. Every element of an endsegment is a finite number however.

"If we think the numbers p/q in such an order [...] then every number
p/q comes at an absolutely fixed position of a simple infinite sequence"
[E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer, Berlin (1932) p. 126]
"The infinite sequence thus defined has the peculiar property to contain
the positive rational numbers completely, and each of them only once at
a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

 

If you accept these claims, then no number must remain in an endsegment.

True: no number is in all endsegments.
False: there is a segment with no numbers.

Numbers within endsegments cannot appear as indices (except one).
Regards, WM