The existence of dark numbers proved by the thinned out harmonic, series

On 18.03.2025 13:18, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

All elements of ℕ are there. That is the assumption. If no greatest can
be identified, then the reason are dark numbers.

 No, the reason is that there is no greatest element.

How can that be realizied? If all are there and all are smaller than omega, tzen there is a greatest one. If there is no greatest one, then the reason is either that new numbers are created (potential infinity) or that dark number become visible. There is no other alternative.

"Dark numbers" do
not exist, as has been proven in this thread.

Dark numbers do exist as has been proven in this thread:
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/ 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 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 of 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.
Unable to comprehend?

"Potential infinity" doesn't exist in modern mathematics.

Therefore modern mathematics is rubbish. By the way this degenerated form of "mathematics" is the first kind which has turned out completely useless for any application. Small wonder. It is wrong.
Regards, WM