# What were the not-so-convincing reasons for using the word “power” for power sets?

A footnote of Enderton's Elements of Set Theory (1977, page 4) for the definition of power set states that

the reasons for using the word "power" in this context are not very convincing, but the usage is now well established.

I wonder what the not-so-convincing reasons were.

EDIT 1: A question regarding the etymology of power is here.

EDIT 2: On page 141, Enderton wrote that the term "power set" is rooted in the fact that card $$\mathscr{P}A$$ equals $$2$$ raised to the power card $$A$$.

• I have no historical knowledge, but I always assumed it was because for a set with $N$ elements, the power set has $2^N$ elements. – Nathaniel Jan 24 at 17:50
• For what it's worth, Zermelo's "Untersuchungen über die Grundlagen der Mengenlehre. I" (Mathematische Annalen, v65, 1908) uses the term "Potenzmenge" (meaning "power set") as if it needed no explanation. The idea but not the term is present in Russell's 1903 Principles of Mathematics ( people.umass.edu/klement/pom/pom.html ). – kimchi lover Jan 24 at 18:38
• Similar to @Nathaniel, I have no factual knowledge. But the function space $f:A\to B$ is often written $B^A$. And then the set of all subsets of $X$ is just (isomorphic to) $2^X$, perhaps suggesting the name "powerset". – John Forkosh Jan 25 at 5:27
• On the other hand, Cantor's Ueber eine elementare Frage der Mannigfaltigkeitslehre (1891), where he proves that the "power" (Mächtigkeit, now cardinality) of the set of subsets is greater than that of the underlying set by the "diagonal argument", does not call the set of subsets "Potenzmenge", or anything else in particular. – Conifold Jan 25 at 5:50
• @JohnForkosh because there are $\lvert B \rvert^\lvert A \rvert$ such functions... – vonbrand Feb 27 at 18:11

As best as I can tell, the term Potenzbegriff (powerclass, later variation Potenzmenge, powerset) was introduced by Bernstein in the late 1890s (Cantor did not use it in his papers). In his Habilitation dissertation Untersuchungen aus der Mengenlehre (1901, published 1905) he states in the introduction (my translation):

"The introduction of the concept of powerclass, and the proof of the equivalence theorem to be mentioned later has now made it possible to arrive at conclusions that were previously only possible through laborious detours by an almost elementary calculation."

The word begriff literally means "concept", and intensional classes (falling under a concept) were not fully separated from extensional sets until Hausdorff's Grundzüge der Mengenlehre (1914), although Zermelo and others operated with the latter earlier. The "equivalence theorem" is now called the Schröder–Bernstein theorem, and Bernstein gives a proof in §1 of the dissertation. The original proof was given at Cantor's seminar in Halle in 1897, and does not survive (Dedekind gave a proof in 1887, but did not publish, Schröder's 1896 announcement was of a proof that apparently had a flaw). The set power (Potenz) is introduced in §2 as follows:

"If $$M$$ and $$N$$ are two sets, we call that set which - in the sense of a known expression - contains all combinations of elements from $$M$$ to the classes of $$N$$, the power $$M^N$$ ($$M$$ raised to $$N$$). Regarding the application to addition, multiplication and powerclasses of commutative and associative laws, they are the same as for finite numbers".

So the apparent motivation for Potenz (power) seems to be as commonly assumed, by analogy to raising numbers to powers. Bernstein does not use $$2^N$$ generally, or calls the class of all subsets the powerclass. But he does write $$2^{\aleph_\alpha}$$, meaning set cardinality (Mächtigkeit, often also translated as "power") in §9, when discussing the continuum hypothesis.

• That's a good observation on the distinction between Potenz and Mächtigkeit that is otherwise obscured in translation. Of course, Mächtigkeit can be more literally translated as "mightiness". I think "power" caught on as a translation of mächtigkeit because it was first translated to puissance in French. – Robert Furber Feb 9 at 17:00
• @RobertFurber "Mächtigkeit" really means more "size" than "power" in German. – vonbrand Feb 27 at 18:14
• @vonbrand I didn't say it meant power. In fact, I said that the conventional translation to English obscured the difference between Potenz and Mächtigkeit. – Robert Furber Feb 28 at 5:56