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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Wed, 6 Nov 2024 16:59:43 +0200
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On 2024-11-05 11:26:58 +0000, WM said:

> On 05.11.2024 11:29, Mikko wrote:
>> On 2024-11-04 18:12:55 +0000, WM said:
> 
>>>> There is no nearesst one. There is always a nearer one.
> 
> And always the endpoint is irrational.
> 
>> That you don't even try to support your clam to support your claim
>> indicates that you don't really believe it.
> 
> My claim says that every point outside of intervals has an irrational 
> interval end next to it. It does not matter how many intervals you 
> claim between the point and the nearest interval, because _all_ 
> intervals have irrational endpoints.
> 
>   Cantor's results are
>> conclusions of proofs and you have not shown any error in the proofs.
> 
> I have. This example for instance proves that he did not enumerate all 
> rationals, because the rationals are dense, the intervals are not dense.

You have not proven that. It is fairly easy to prove that there are
no positive rationals other than those enumerated by Cantor (if I
recall correctly he enumerated only positive rationals). To prove
that there are positive rationals that are not included in Cantor's
enumeration it suffices to show one but you have not shown any.

>> You are free to deny one of more of the assumptions that constitue
>> the foudations of the results but you havn't.
> 
> Cantor's bijections concern only potentially infinite sets, but are 
> assumed and claimed to concern the complete sets.

Everything Cantor said was about complete sets. He did neither deny the
possibility of potentially infinte sets nor said anything about them (as
far as I know and remember).

> That is the grave mistake. His result says for all infinite "countable 
> sets" that they are infinite, nothing more.

He very clearly says and proves that all infinite sets are not equinumerous.

-- 
Mikko