This question asks for the following deviation from Cantor's proof: Cantor enumerated the positive fractions by the natural numbers n. Mueckenheim enumerates the positive fractions by the integer fractions n/1. Since these integer fractions also have to be enumerated, the number of not enumerated fractions never decreases. But all definable fractions get enumerated.
Is the use of integer fractions a mistake?
According to Mueckenheim's proof of dark numbers countability is fake. I cannot find anything wrong. On the other hand it seems very strange.
Recently Mueckenheim has published in sci.math three proofs of dark numbers. For convenience I simply copy here the first and most convincing one.
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 simply 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. They cannot be enumerated.