Re: The existence of dark numbers proven by the thinned out harmonic series

On 12.03.2025 11:22, Alan Mackenzie wrote:

WM <invalid@no.org> wrote:

What I think you're trying to say is that the sum of all terms
containing both 8 and 9 diverges.  It is far from clear that this is the
case.

It is a simple fact. I recommend that you learn it from my lecture
https://www.hs-augsburg.de/~mueckenh/HI/HI02 p.15.

 

We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8,
9, 0  in the denominator without changing this.

 What do you mean by "this"?  What does it refer to?

Kempner has proved that the series S(9) which contains all fractions without digit 9 is converging. That implies that the remaining series S-S(9) is diverging. We ca also prove that the series S(8) which contains all fractions without digit 8 converges. That implies that the remaining series is diverging. Of course also S(8) subtracted from S-S(9) is converging, and the remainig series S-S(9)-S(8) is diverging. And so on. Only terms with digits 8 and 9 simultaneously belong to the series S-S(9)-S(8).

 

That means that only the terms containing all these digits together
constitute the diverging series. (*)

 There are many diverging sub-series possible.  I think you mean "a"
diverging series.

No, I means all of them: S-S(9)-S(8)-...-S(0) contains only terms which have digits 1234567890 in any desired order.

 

But that's not the end! We can remove any number, like 2025, ....

  From what?  2025 isn't a digit.

Use base 2026. Then 2025 is a digit.

 

.... and the remaining series will converge. For proof use base 2026.
This extends to every definable number.

 Meaningless.  "Definable number" is itself undefined.

Definition: A natural number is "named" or "addressed" or "identified" or "(individually) defined" or "instantiated" if it can be communicated, necessarily by a finite amount of information, in the sense of Poincaré[1], such that sender and receiver understand the same and can link it by a finite initial segment (1, 2, 3, ..., n) of natural numbers to the origin 0. All other natural numbers are called dark natural numbers.
  Communication can occur
  - by direct description in the unary system like ||||||| or as many beeps, raps, or flashes,
  - by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7),
  - as n-ary representation, for instance binary 111 or decimal 7,
  - by indirect description like "the number of colours of the rainbow",
  - by other words known to sender and receiver like "seven".
  [1] "In my opinion a subject is only conceivable if it can be defined by a finite number of words."
Regards, WM