New proof of dark numbers by means of the thinned out harmonic series

Wolfgang Mückenheim
Technische Hochschule Augsburg
wolfgang.mueckenheim@tha.de


Abstract: It is shown by the intersection of the complements of all
Kempner series belonging to definable natural numbers that not all
natural numbers can be defined. We call them dark.


1. Dark Numbers

Not all numbers can be chosen, expressed, and communicated as
individuals such that the receiver knows what the sender has meant. We
call those numbers dark numbers. Much evidence has been collected and
discussed [1]. But in the following we will present the shortest proof
of their existence. Of course the facilities to express numbers depend
on the environment and the power of the applied system. But this proof
shows that, independent of the system, infinitely many natural numbers
will remain dark forever.

A simple example is provided already by the denominators of the harmonic
series (1/n)n∈. Whatever attempts are made to express denominators m as large as possible, the sum from 1/1 to 1/m is finite while the remaining
part of the series diverges.


2. Kempner Series

The harmonic series diverges. But as Kempner [2] has shown in 1914,
deleting all terms containing the digit 9 turns it into a converging
series, the Kempner series, here abbreviated as K(9). That means that
the complement C(9) of removed terms

,

all containing the digit 9, carries the divergence alone. All other
terms can be removed. Same is true when all terms containing the digit 8
are removed. That means that the complement C(8)

.

of the Kempner series K(8) carries the divergence alone. Since here
those terms containing the digits 9 without digits 8 belong to the
converging series K(8) we can conclude that the divergence is caused by
the intersection only, i.e., by all terms containing the digits 8 and 9 simultaneously:


3. Proof

But not all terms containing 8 and 9 are needed. We can continue and
remove all terms containing 1, 2, 3, 4, 5, 6, 7, or 0 in the denominator without changing this result because the ten corresponding Kempner
series K(0), K(1), ..., K(9) are converging and their complements C(0),
C(1), ..., C(9) are diverging. But only the intersection of all
complements carries the divergence. That means that only the terms
containing all the digits 0 to 9 simultaneously constitute the diverging series.

But that is not the end! We can remove any natural number k, like 2025,
and the remaining Kempner series will converge. For proof use base 2026
where 2025 is a digit. This extends to every defined number, i.e., every
number k that can be defined, chosen, and communicated such that the
receiver knows what the sender has meant. When the terms containing k
are deleted, then the remaining series converges.


4. Result

The diverging part of the harmonic series is constituted only by
intersection of all complements C(k) of Kempner series K(k) of defined
natural numbers k, i.e., by all the terms containing the digit sequences
of all defined natural numbers. No defined natural number exists which
must be left out. Terms which, although being larger than every defined
number, do not contain all defined digit sequences, for instance not
Ramsey's number, belong to converging Kempner series and not to the
diverging series of the intersection of all complements. All infinitely
many terms containing not the digit 1 or not the digit sequence 2025 or
not the digit sequence of Ramsey's number can be deleted without
violating the divergence.

All Kempner series K(k) of defined, i.e., finite numbers k split off in
this way are converging and therefore the sum of their always finite
sums is finite too although it can be very large [3]. The divergence
however remains. It is carried only by terms which are dark and greater
than all digit sequences of all defined numbers  we can even say
greater than all digit sequences of all definable numbers because, when
larger numbers will be defined in future, they will behave in the same
way. It is impossible to choose a natural number such that the
intersection of the complements of all Kempner series of larger numbers
is finite.

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


Literature

[1] W. Mückenheim: "Evidence for Dark Numbers", ELIVA Press, Chisinau,
2024, pp. 1-36.
[2] A. J. Kempner: "A Curious Convergent Series", American Mathematical
Monthly 21 (2), 1914, pp. 48–50.
[3] T. Schmelzer, R. Baillie: "Summing a Curious, Slowly Convergent
Series", American Mathematical Monthly 115 (6), 2008, pp. 525–540.

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