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Cross posted at Math Overflow

For historical reasons, the English term "power set" in set theory is a translation of the German "Potenzmenge", which is still in use in German mathematical literature.

Did some English language mathematics use the term "potency set" for power set, and if so, what are good references.

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    $\begingroup$ Previously asked in mathoverflow.net/questions/457909/potency-set-for-power-set ... there, the conclusion was that "potency set" was never used in English. $\endgroup$ Nov 14, 2023 at 17:38
  • $\begingroup$ @GeraldEdgar It is indeed true that the question was asked there. But the participants here are different, and the fact that no-one there found a use does not show that the term was never used in English. $\endgroup$ Nov 14, 2023 at 18:18
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    $\begingroup$ Re, I think that @GeraldEdgar did not mean to stymie discussion here, only to make participants here (not all, but not none, of whom are also participants at MO) aware of existing discussion. It is generally appropriate to edit a link to a cross post into both questions; I have edited your MO question accordingly. $\endgroup$
    – LSpice
    Nov 15, 2023 at 18:50
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    $\begingroup$ @LSpice Thanks for the clarification and the edit. $\endgroup$ Nov 15, 2023 at 22:02

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The paper Rieger, L. "On the consistency of the generalized continuum hypothesis." Rozprawy Mat. 31 (1963) described in MR0146091, which can be downloaded from http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-5ca4b532-4aa3-4856-924f-8bbed25e3c33 (click on the "31. książka") uses the phrase "potency set axiom" on pp 15, 20, 38 (and maybe elsewhere).

When discussing the "potency set axiom" Rieger refers to axiom C3 in Gödel's 1940 The Consistency of the Continuum Hypothesis (Annals of Mathematics Studies, Princeton), whose text is reproduced in vol II of the 1990 edition of Gödel's Collected Works. Towards the end of Chapter II (on p.32 of the Collected Works) is a group of 4 axioms, labeled "Group C". Gödel describes the third of them as providing for the existence of a "set including the set of all subsets" of a given set, but does not use any of the terms "Potenzmenge", "power set" or "potency set".

So I suppose Rieger translated the usual term "Axiom der Potenzmenge" (used by Zermelo in his 1907 paper) into English, with the help of a dictionary, not knowing that "power set" is idiomatic and that "potency set" is never used. (See this for info about Rieger's life.)

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    $\begingroup$ @FrodeAlfsonBjørdal Ladislav Svante Rieger. According to Math Genealogy he had 73 descendants. His degree was from Charles University and his first publications were in Czech journals, so I doubt English was his first language. $\endgroup$ Nov 14, 2023 at 22:10
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    $\begingroup$ One might add that Reiger's paper was published in a Polish journal/bookborsu, where a mistranslation to English might not have been caught. $\endgroup$
    – Michael E2
    Nov 15, 2023 at 3:03
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    $\begingroup$ @FrodeAlfsonBjørdal A more literal translation seems a mistranslation to me, since "power set" is jargon. Like translating "corps locaux" as "local bodies" instead of "local fields." And "power" is, according to my dictionary, is a more common translation of "Potenz." "Potency" is not a commonly used word in English, except in "drug potency," and much less common than power or potential. $\endgroup$
    – Michael E2
    Nov 15, 2023 at 12:43
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    $\begingroup$ MR0106832 (of Wegel, Herbert Axiomatische Mengenlehre ohne Elemente von Mengen., Math. Ann. 131 (1956)) is by L. Rieger, who says near the end of his review "the power of the potency-set", meaning I am sure "the cardinality of the power set". $\endgroup$ Nov 16, 2023 at 14:19
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    $\begingroup$ @FrodeAlfsonBjørdal You are right, but I am right, too. The article appeared in Band 65, Heft 1, dated 13 December 1907, as seen on gdz.sub.uni-goettingen.de/id/… . If you skip to the end of the article, you see it was submitted on 30 July 1907, as shown on gdz.sub.uni-goettingen.de/id/… . When the last Heft of volume 65 appeared it was 1908. $\endgroup$ Nov 16, 2023 at 18:25

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