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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Thu, 19 Dec 2024 15:38:59 +0100
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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