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Ordinal multiplication has always seemed backwards to me. $\alpha \times \beta$ is defined as the concatenation of $\beta$ copies of $\alpha$, not the other way 'round as one might expect.

Does this go back to Cantor? Or did someone else create this convention?

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  • $\begingroup$ I hesitate to edit the title myself, but it might be more precisely to the point to ask about the convention for ordinal mult, rather than just "definition"... which connotes something different. $\endgroup$
    – paul garrett
    Commented Jan 26, 2019 at 23:16
  • $\begingroup$ @paulgarrett Yes I can see that. But isn't the convention the definition? I'm not sure what the difference is. Unless you mean that the two possible conventions are essentially the same definition, as opposed to some completely different definition. I don't care if you want to edit it. A few weeks ago someone suggested adding some tags and I rejected the edit by mistake, if that person wants to try again I'll approve. $\endgroup$
    – user4894
    Commented Jan 26, 2019 at 23:26
  • $\begingroup$ Yes, I did mean that the two different conventions really do "define" the same thing, it's just a notational thing, not mathematical. I'll edit your title, then, if you don't object. $\endgroup$
    – paul garrett
    Commented Jan 26, 2019 at 23:45
  • $\begingroup$ @paulgarrett You're right, it's the same definition. Different convention. You convinced me! $\endgroup$
    – user4894
    Commented Jan 27, 2019 at 4:25

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Cantor at first defined multiplication of ordinals one way, then later (for what he thought were good reasons) switched to the opposite way. So YES it is due to Cantor.

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  • $\begingroup$ A citation would be helpful. $\endgroup$
    – Carl Witthoft
    Commented Jun 18, 2018 at 13:11
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    $\begingroup$ Citation? All I can say now (many years later) is that I learned this from Dauben's biography amazon.com/Georg-Cantor-Mathematics-Philosophy-Infinite/dp/… $\endgroup$
    – Gerald Edgar
    Commented Jun 18, 2018 at 13:18
  • $\begingroup$ The relevant passage from Dauben's book is note 46 on page 345: “Similar results were expressed somewhat differently in the Grundlagen because there the order of multiplication was reversed from the form adopted in the Beiträge. [...] Cantor made specific reference to this change at the end of Section 17 of the Beiträge [...]” $\endgroup$
    – Hans Lundmark
    Commented Apr 21, 2022 at 6:57

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