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From: "Fred. Zwarts" <F.Zwarts@HetNet.nl>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Tue, 8 Apr 2025 21:22:49 +0200
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Op 08.apr.2025 om 20:44 schreef Andy Walker:
> On 08/04/2025 16:17, Richard Heathfield wrote:
>> It will, however, take me some extraordinarily convincing
>> mathematics before I'll be ready to accept that 1/3 is irrational.
> 
>      I don't think that's quite what Wij is claiming.  He thinks,
> rather, that 0.333... is different from 1/3.  No matter how far you
> pursue that sequence, you have a number that is slightly less than
> 1/3.  In real analysis, the limit is 1/3 exactly.  In Wij-analysis,
> limits don't exist [as I understand it], because he doesn't accept
> that there are no infinitesimals.  It's like those who dispute that
> 0.999... == 1 [exactly], and when challenged to produce a number
> between 0.999... and 1, produce 0.999...5.  They have a point, as
> the Archimedean axiom is not one of the things that gets mentioned
> much at school or in many undergrad courses, and it seems like an
> arbitrary and unnecessary addition to the rules.  But we have no good
> and widely-known notation for what can follow a "...", so the Wijs of
> this world get mocked.  He doesn't help himself by refusing to learn
> about the existing non-standard systems.
> 

To me it seems that it comes down to the definition of real numbers.
One definition is in terms of limits of a series of numbers. Once one 
understands the definition of limits, it is clear that different series 
can be used for the same real number. The real number 1, e.g., can be 
defined by (amongst others) the following series:
0.9, 0.99, 0.999, 0.9999, ...
1, 1, 1, 1, ....
1.1, 1.01, 1.001, 1.0001, ...
Two series X(n) and Y(n) indicate the same real number if for each small 
ε one can find a number N so that for all n>N |X(n)-Y(n)| < ε.
(See https://en.wikipedia.org/wiki/Construction_of_the_real_numbers)
Apparently the Wij-analysis is not about real numbers, but it is not 
clear what alternative numbers are used.