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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Tue, 27 May 2025 17:18:39 +0200
Organization: A noiseless patient Spider
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On 27.05.2025 14:15, Mikko wrote:
> On 2025-05-26 23:21:27 +0000, Ben Bacarisse said:

>> Mind you, all I know if the ambition of the institution in question is
>> that it allows nonsense to be taught, albeit as a entirely optional
>> course.  That particular hochschule does not seem to be taking academic
>> integrity very seriously.
> 
> Maybe the idea is that an exposure to nonsense helps students to learn
> to identify nonsense when they see it.
> 
There is no nonsense but most mathematicians are unable to understand my 
arguments and therefore do not answer them but simply curse.

The harmonic series diverges. Kempner has shown in 1914 that when all 
terms containing the digit 9 are removed, the series converges. Here is 
a simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/HI02.PPT, 
p. 15.

That means that the terms containing 9 diverge. Same is true when all 
terms containing 8 are removed. That means all terms containing 8 and 9 
simultaneously diverge.

We can continue and remove all terms containing 0, 1, 2, 3, 4, 5, 6, 7, 
8, 9 in the denominator without changing this. That means that only the 
terms containing all these digits together constitute the diverging series.

But that's not the end! We can remove any number, like 2025, and the 
remaining series will converge. For proof use base 2026. This extends to 
every definable number. Therefore the diverging part of the harmonic 
series is constituted only by terms containing a digit sequence 
including all definable numbers.

Note that here not only the first terms are cut off but that many 
following terms are excluded from the diverging remainder.

This is a proof of the huge set of undefinable or dark numbers.

Regards, WM