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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Fri, 13 Dec 2024 11:42:13 -0800
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On 12/13/2024 2:55 AM, WM wrote:
> On 13.12.2024 03:23, Richard Damon wrote:
>> On 12/12/24 9:25 AM, WM wrote:
>>> if Cantor can apply all natural numbers as indices for his 
>>> bijections, then all must leave the sequence of endsegments. Then the 
>>> sequence (E(k)) must end up empty. And there must be a continuous 
>>> staircase from E(k) to the empty set.
>>>
>> But a segment that is infinite in length is, by definiton, missing at 
>> least on end.
> 
> That means that the premise "if Cantor can apply all natural numbers as 
> indices for his bijections" is false.

Of course cantor pairing can be indexed. You just don't know. Whatever.


>> So, which bijection from Cantor are you talking about? Of are you 
>> working on a straw man that Cantor never talked about?
> 
> There are many. The mapping from natumbers to the rationals, for 
> instance, needs all natural numbers. That means all must leave the 
> endsegments. Another example is Cantor's list "proving" uncountable 
> sets. If not every natural number has left the endsegment and is applied 
> as an index of a line of the list, the list is useless.
> 
> But if every natural number has left the endsegments, then the 
> intersection of all endsegments is empty. Then the infinite sequence of 
> endegments has a last term (and many finite predecessors, because of
> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}).
> 
> Regards, WM
> 
> 
>