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

On 18.12.2024 21:15, joes wrote:

Am Wed, 18 Dec 2024 20:06:19 +0100 schrieb WM:

On 18.12.2024 13:29, Richard Damon wrote:

On 12/17/24 4:57 PM, WM wrote:

>

You claimed that he uses more than I do, namely all natural numbers.

Right, you never use ALL the natural numbers, only a finite subset of
them.

Please give the quote from which you obtain a difference between "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)] and
my "the infinite sequence f(n) = [1, n] contains all natural numbers n
completely, and each of them only once at a determined place."

You deny the limit.
 

When dealing with Cantor's mappings between infinite sets, it is argued usually that these mappings require a "limit" to be completed or that they cannot be completed. Such arguing has to be rejected flatly. For this reason some of Cantor's statements are quoted below.
"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]
"thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the th algebraic number where not a single one of this epitome () has been forgotten." [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 116]
"such that every element of the set stands at a definite position of this sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152]
The clarity of these expressions is noteworthy: all and every, completely, at an absolutely fixed position, th number, where not a single one has been forgotten.
Regards, WM