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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Thu, 14 Nov 2024 11:20:33 +0100
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On 14.11.2024 00:16, Jim Burns wrote:

> A finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 in which
> each claim is true.or.not.first.false
> is
> a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 in which
> each claim is true.
> 
> Some claims are true and we know it
> because
> they claim that
> when we say this, we mean that,
> and we, conscious of our own minds, know that
> when we say this, we mean that.
> 
> Some 𝗰𝗹𝗮𝗶𝗺𝘀 are not.first.false and we know it
> because
> we can see that
> no assignment of truth.values exists
> in which 𝘁𝗵𝗲𝘆 are first.false.
> 𝗾 is not first.false in ⟨ 𝗽 𝗽⇒𝗾 𝗾 ⟩.
> 
> Some finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲𝘀 of 𝗰𝗹𝗮𝗶𝗺𝘀 are
> each true.or.not.first.false
> and we know it.
> 
> When we know that,
> we know each claim is true.
> 
> We know each claim is true, even if
> it is a claim physically impossible to check,
> like it would be physically impossible
> to check each one of infinitely.many.
> 
Here is a single claim which is true:

The covering of a geometric figure by a set of similar smaller intervals 
is independent of the order of the intervals. That holds for every 
finite figure and, by applying the analytical limit, also for infinite 
figures like

XOOO...
XOOO...
XOOO...
XOOO...
....

or

0---------_1_--------_2_--------_3_---...

Therefore a geometric representation let alone proof of most of Cantor's 
bijections is impossible.

Belief in set theory excludes belief in geometry.

All babble about Completely Scattered Space in cases of
I(n) = [n - sqrt(2)/2^n, n + sqrt(2)/2^n]
or
I(n) = [n - 1/2^n, n + 1/2^n]
is to no avail and useless from the outset.

Regards, WM