Why was the cardinal number of a set also called its "power"? ( Reference : Cantor)

Question

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?

Answer 1

"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)".

Answer 2

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.