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Mueckenheim published in sci.math three proofs of dark numbers. For convenience I simply copy them here. Are they valid?

(1) Cantor has proved that all positive fractions m/n can be enumerated by all natural numbers k:

k = (m + n - 1)(m + n - 2)/2 + m. (*)

This is tantamount to enumerating the positive fractions by the integer fractions of the first column of the matrix

1/1, 1/2, 1/3, 1/4, ...

2/1, 2/2, 2/3, 2/4, ...

3/1, 3/2, 3/3, 3/4, ...

4/1, 4/2, 4/3, 4/4, ...

...

Of course also the integer fractions belong to the fractions to be enumerated. Therefore his approach is tantamount to exchanging X's and O's in the matrix until all O's have disappeared:

X, O, O, O, ...

X, O, O, O, ...

X, O, O, O, ...

X, O, O, O, ...

...

In fact by application of (*) all O's are removed from all visible or definable matrix positions. However it is clear that, by simple exchanging O's with X's, never an O will be removed from the matrix. This shows that the O's move to invisible, i.e., undefinable matrix positions. These are called dark positions.

(2) The intersection of non-empty inclusion-monotonic sets like infinite endsegments E(k) = {k, k+1, k+2, ...} is not empty. Every non-empty endsegment shares at least one natural number with all non-empty endsegments. In fact every infinite endsegment shares infinitely many natural numbers with all infinite endsegments. Otherwise there would be a first endsegment sharing less natural numbers with its predecessors. This cannot happen, if all endsegments are infinite.

But according to ZFC, the intersection of all endsegments is empty. Since all definable endsegments satisfy

∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀

the empty intersection cannot be accomplished by merely definable endsegments

∩{E(k) : k ∈ ℕ_def} =/= { }.

Only by the presence of undefinable endsegments

∩{E(k) : k ∈ ℕ} = { }

can be accomplished.

(3) The simplest proof of dark natural numbers is this:

Every definable natural number k is finite and belongs to a finite set

{1, 2, 3, ..., k}.

If there are ℵo, i.e., more than any finite number, then ℕ can can only be filled and completed by dark natural numbers. This is obvious from the simple fact

∀k ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., k}| = ℵo .

Regards, WM

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  • Could you make more precise what you mean by "tantamount to exchanging X's and O's" and "undefinable matrix positions"?
    – Ben McKay
    4 hours ago
  • When an "index" k/1 is applied to a fraction m/n, then the fraction k/1 remains without index. So the indeX is moved to m/n and the nO-index is moved to k/1. 4 hours ago  
  • The sentence "This shows that the O's move to invisible, i.e., undefinable matrix positions" is clearly mistaken. Sequences do not always have limits, even if we try to "attach points at infinity". To declare them to have "undefinable limits" is just to use meaningless language, and then we might as well give up language altogether. So I suggest giving up language altogether when communicating with the author of these "dark numbers".
    – Ben McKay
    2 hours ago
  • Fact is, the O's cannot leave the matrix. The problem is not a limit because Cantor's enumeration of the rationals is not a limit-process either. Every rational must be enumerated. If it was only correct in the limit, no-one would have believed him. 2 hours ago  
  • Your description of moves of Xs and Os is a description of a sequence of moves applied to a matrix, i.e. a sequence of matrices. The question is then where each Os goes in this sequence. Each O moves in a sequence of matrix positions, without a limit. Cantor does not need limits, but the question of where an O moves to is a question of limits of that sequence. If that is not how you interpret the question, you will want to make a mathematically precise definition of the question you are asking.
    – Ben McKay
    2 hours ago
  • Cantor does not need limits. That is correct. And Mueckenheims X's seem to model precisely Cantor's formula. The O's have never been objects of attention. 2 hours ago  
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"Under Cherries" is an obvious sock puppet of Pr. Wolfgang Mückenheim, who is unfortunately "teaching" this kind of idiocies (amongst many others) to real students at Hochschule Augsburg. This is a shame for german Academy.

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  • Good to know, but sadly not an answer. Anyway, with a score currently of -9 votes, the question will get deleted, and with it this answer, too.
    – Alex M.
    16 mins ago
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