Who was the first to mention the nine problem?

Question

When Cantor published his seminal diagonal argument, he used binary code and did not distinguish the real numbers 0.1000… and 0.0111… Later, Koenig found a way to save the diagonal argument in binary digits (a reference would be appreciated). Meanwhile, the most popular form is to use decimal digits.

My main question: Who was the first person to mention that in the diagonal argument digits should not be replaced by 9's because of 0.1999⋯ = 0.2000… ?

Answer 1

The first mention of the "nine-problem" was by Felix Klein, Vorträge über ausgewählte Fragen der Elementargeometrie, Teubner 1895, p. 42. http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACV2370.0001.001

Answer 2

When Cantor published his seminal diagonal argument, he used binary code and did not distinguish the real numbers 0.1000… and 0.0111…

This is not relevant for the proof.

See :

• Joseph Dauben, Georg Cantor : His Mathematics and Philosophy of the Infinite (1990)

and

• José Ferreiros, Labyrinth of thought : A history of set theory and its role in modern mathematics (2ed 2007)

For detailed discussion of Cantor's proofs:

1873 proof: Dauben page 50 and Ferreiros page 179.

1877 proof: Ferreiros page 190-2. See page 190, for the presumibely source of the misconception.

1891 proof (the diagonalization one): Dauben page 165 and Ferreiros page 287.

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For an early textbook version of the diagonal proof, see:

• William Henry Young & Grace Chisholm Young, The Theory of Sets of Points, (1906) page 43 : decimal notation used, and the rule for "diagonalization" is :

"we now define a number [...] $b=0.b_1b_2b_3 \ldots$, where $b_r$ is never the same as $a_{rr}$."

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Regarding the purported "mistake" in the diagonal proof, see e.g. Robert Gray Georg Cantor and Transcendental Numbers, The American Mathematical Monthly, vol. 101, 1994, page 823.

The "general" theorem consider an enumeration of an infinite sequnce of infinite strings of symbols, with only two symbols : $m$ or $w$.

The proof does not rely on any property of real numbers (or their representation).

It is easily "applied" to the real numbers through their binary representation.

In the case of number with a "dual" representation, both representation must be included into the enumeration.

We can consider a simplified case :

$$a_1=0.000 \ldots$$

$$a_2=0.011 \ldots$$

$$a_3=0.100 \ldots$$

$$\ldots$$

and apply Young's procedure; we will get :

$$b=0.101 \ldots$$

because: $b_1 \ne a_{11}=0$, $b_2 \ne a_{22}=1$, $b_3 \ne a_{33}=0$.

The presence of $a_2$ and $a_3$, that are two "dual" representation of the same number, does not invalidate the diagonalization.

With this simple example we see that also if two members of the list, say $a_n$ and $a_m$, are two distinct representations of the same number we have that :

$b \ne a_n$, because $b_n \ne a_{nn}$

and :

$b \ne a_m$, because $b_m \ne a_{mm}$

and this "works" irrespectively of the fact that $a_n=a_m$ or not.