Re: Cantor Diagonal Proof

On 4/8/25 3:22 PM, Fred. Zwarts wrote:

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.
 

Yes, that becomes the key issue, what is the definition, and all the definitions (at least generally accepted) make the Real Numbers all "Finite", excluding trans-finite numbers on either the infintesimal or the infinite variety (or even those between the reals).
Wij just refuses to accept that definition, which just breaks his ability to communcate his ideas. I suppose that part of the problem is that much of his ideas are old-hat once you accept that we are talking in a beyond-the-reals number system.