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I had a discussion with one of my students, who was convinced that they could prove something was countable using Cantor's diagonal argument. They were referring to (what I know as) Cantor's pairing function, where one snakes through a table by enumerating all finite diagonals, e.g. to prove the countability of $\Bbb N\times\Bbb N$. In the same way one proves that $\Bbb Q$ is countable.

I was therefore very surprised to learn that this application of the pairing function is titled "Cantor's first diagonal argument" on the German Wikipedia. Is this title for Cantor's proof of the countability of $\Bbb Q$ in any way standard (in German), i.e., is it mentioned by this title in any literature outside of the German Wikipedia?

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    $\begingroup$ Google returns several instances: google.com/search?q=Cantor%27s+first+diagonal+argument. Personally, I find this bad terminology, since it invites a confusion with true diagonalization, a totally different idea. $\endgroup$
    – Joel David Hamkins
    Commented Jul 13, 2023 at 14:35
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    $\begingroup$ @JoelDavidHamkins How likely is it that these Google results were created because the authors read the Wikipedia article in the first place? (AKA citogenesis: xkcd.com/978) $\endgroup$
    – Najib Idrissi
    Commented Jul 13, 2023 at 14:36
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    $\begingroup$ I honestly doubt the terminology is standard. Someone asked in the Wikipedia article's discussion page, in 2007 (!), whether anyone had any source that the argument is really called the diagonal argument. The only reference found was in a paper of Wittgenstein (who was a philosopher of mathematics, not a mathematician). He apparently didn't even call it the diagonal argument but the diagonal method. And I can't find the text of Bemerkungen über die Grundlagen der Mathematik Teil II to even confirm he actually called it that. $\endgroup$
    – Najib Idrissi
    Commented Jul 13, 2023 at 14:38
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    $\begingroup$ When I learned about Cantor's work (probably around 1960), "first" and "second" diagonal argument was the terminology I learned (in English). $\endgroup$
    – Andreas Blass
    Commented Jul 13, 2023 at 14:56
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    $\begingroup$ I have heard the terminology since my freshman year, in multiple countries, and often online from multiple sources. For the most part this is done by non-logicians and then gets parroted through the years over lecture notes or oblivious students helping or private tutoring. I agree with the voices here, this has always irked me, and it is a terrible and confusing terminology. $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 14, 2023 at 14:18

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