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Below, a page of Cantor's Contributions To The Founding Of The Theory Of Transfinite Numbers ( at archive.org) Is there a mathematical analogy explaining why the magnitude of a set can be called its " power"?

Aside : which German word does " power " translate?

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Andrés E. Caicedo
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    Maybe it sounded good in German? – Angina Seng Dec 01 '19 at 18:20
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    Whether this question belongs on math.se or hsm.se may depend on whether this term has been used much since Cantor (as far as my own experience goes, it hasn't). – J.G. Dec 01 '19 at 18:25
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    @J.G. The term seems not to be used, but " equipotent' is still used occasionaly I think. –  Dec 01 '19 at 18:27
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    Whether that's due to "power" or equipollent seems relevant, but I'm not sure which. – J.G. Dec 01 '19 at 18:31
  • According to a latin dictionary " polleo" means " having much power" , so the idea seems to be the same in "equipotent" and in " equipollent". –  Dec 01 '19 at 18:34
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    Re the aside: Mächtigkeit. – Daniel Fischer Dec 01 '19 at 18:37
  • @DanielFischer. Thanks. –  Dec 01 '19 at 18:40
  • @DanielFischer. According to you, does " Machtigkeit" connote strength or rather potentiality ( as opposed to actuality). ( Sorry, do not know how to type the lettre " a" with an Umlaut). –  Dec 01 '19 at 18:45
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    Uh, that's difficult. "Mächtigkeit" has many meanings, one of which is "largeness" (which can explain why that word was used in German). Others relate more closely to "power". Connotation changes for different meanings. Note that "Mächitgkeit" is derived from "Macht" (power, it's the same word as the noun "might") via the adjective "mächtig", from which then another noun is formed, with related but different meanings. Playing fast and loose with English, it should rather be translated "powerfulness" than "power". – Daniel Fischer Dec 01 '19 at 19:03
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    See also mächtig – Henno Brandsma Dec 01 '19 at 19:10
  • DWDS knows “Mächtigkeit” only as size. And to me (as German) that makes sense; as noun for the “power” meaning of “mächtig” I would use “Macht”, not “Mächtigkeit”. – celtschk Dec 01 '19 at 20:35
  • @celtschk same in Dutch, macht, machtig, machtigheid (noun, derived adjective), noun from that adjective again is "double" so has a specialised meaning. "machtigheid" could also refer to "food" being "heavy". – Henno Brandsma Dec 03 '19 at 08:07

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"Mächtigkeit" in German (and cognates in other Germanic languages) can mean be used in expressions to there are equal numbers: "wy binne like machtich" (West Frisian) "we have the same number of people", which in some situations can imply you're equally "powerful", "Macht" means "power".

So it seems that we have a transfer of meaning from "power" to "number (of people)".

Henno Brandsma
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I suspect power was derived from the French word puissance, which Mittag-Leffler used for Cantor's word Mächtigkeit in Mittag-Leffler's French translations of some of Cantor's work in early issues of Acta Mathematica (Volumes 2 and 4). Also, Borel used puissance in his influential 1898 book Leçons sur la Théorie des Fonctions. The word power was used in a lot of older mathematical literature written in English, indeed even up through the 1950s and later, as well as the word potency. See also footnote * on p. 75 of Huntington's The Continuum and Other Types of Serial Order (2nd edition), where he mentions that Cantor used the word cardinalzahl in 1887.

Une Contribution a la Théorie des Ensembles (see first page, line 5)

Sur les ensembles infinis et linéaires de points (see middle of p. 352)

Fondements d'une théorie générale des ensembles (see p. 384, line 6)

Sur divers théorèmes de la théorie des ensembles de points situes dans un espace continu a N dimensions: Première communication Extrait d'une lettre adressée à l'éditeur (see first page, line 9)

De la puissance des ensembles parfaits de points: Extrait d’une lettre adressée à l’éditeur

Abraham A. Fraenkel writes the following on p. 68 (lines 12-15) of the 1961 2nd edition of Abstract Set Theory:

Cantor ever attempted to accomplish it [= proof of comparability of cardinals] and never succeeded; this is why he used the more neutral term "Mächtigkeit" (power, puissance) rather than the term "cardinal" which should entail comparability.

Dave L. Renfro
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