# What notation, if any, did Principia use for power set?

I have read in Florian Cajori, A history of Mathematical Notation (https://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf) to check notations for the power set notation.

Interestingly, the book does not include Cantor, and it does not attribute notation for power set to Principia Mathematica. The latter was surprising, but it may for all I know be that Russell & Whitehead did not use the power set.

If Principia used notation for power set, what was it?

• @njuffa Russell and Whitehead, the authors of Principia Mathematica. I did not state that Cajori discusses power set notation, and I think that he did not. Commented Aug 5 at 21:45
• @njuffa I have read some in Principia Mathematica, and will do again if no-one shares an answer on the power set issue. Yes, Whitehead was listed as the first author of PM, and I used the alphabetical order here. Commented Aug 5 at 22:56
• Not so surprising IMO. Cajori's History has been published in 1928-29. For sure, such a huge work needed several years to be written. Commented Aug 6 at 15:00

SEP has an article on The Notation in Principia Mathematica by Linsky. $$*60\cdot 12$$ is the notation for power sets $$\vdash.Cl'\alpha=\hat{\beta}(\beta\subset\alpha)$$ and it captures the "classes of subclasses of $$\alpha$$". It's not clearly a specialized notation for power set, rather it's a subclass notation $$'\alpha$$ in conjuction with a "class of" notation $$Cl$$ (compare $$*60\cdot 13$$ for another use of these operators for a different sub class construction).

The coinage of a "Potenzmenge" ("power set") very likely is original to Zermelo (1908) who incidentally uses the notation $$\mathfrak{U}T$$ where $$\frak{U}$$ stands for the German "Untermenge" i.e. subset.

P.S. Added detail from comment exchange below:

Adoption of Zermelo's coinage of Potenzmenge was quite rapid. Hessenberg wrote an extensive Volume on set theory using pre-Zermelo notions (Teilmengen instead of Untermengen, and no mention of Potenzmengen):

However a year after Zermelo's 1908 paper he had already adopted both the $$\mathfrak{U}T$$ notation and the term "Potenzmenge" (power set):

• Many thanks! I accept your answer on account of its useful link to the article by Linsky. :) Confer my comments on the other question hsm.stackexchange.com/questions/17782/…, where I discovered the result by myself, in a roundabout way depending upon Linsky, Bernard and Andrew David Irvine, "Principia Mathematica", The Stanford Encyclopedia of Philosophy (Fall 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), forthcoming URL = <plato.stanford.edu/archives/fall2024/entries/…>. Commented Aug 6 at 15:14
• @FrodeAlfsonBjørdal It's p.265 Axiom IV. The phrase appears twice. Once in description under quotes and one to name the axiom in parethesis below it. Happy to translate the whole passage if that is helpful. Commented Aug 6 at 16:03
• Ah I see! Yeah, Cantor does not use the verbage. The speed of adoption of Zermelo's language is actually remarkable. Hessenberg wrote a text in 1906 and doesn't yet use the term but already in 1909 right after Zermelo's article appeared Hessenberg adopted the terminology. Hessenberg's 1906 source is a good example of pre-Zermelo discussion of Cantor's theorem. Commented Aug 6 at 16:17
• Zermelo splits the difference here. He introduces the word Potenzmenge, but uses the fraktur (that's the word for the german script style) U for Untermenge (which means subset). This is to both connect to the previously used used Teilmenge (the old word for subset in Hessenberg). Problem with U is of course it doesn't translate into English and other languages with the same meaning, so my best guess is that P just ended up making more sense in the non-German literature. I haven't looked that that transition in any detail but my starting guess would be Bourbaki. Commented Aug 6 at 17:12
• @FrodeAlfsonBjørdal Full citation is: Gerhard Hessenberg "Kettentheorie und Wahlordnung" Journal für die reine und angewandte Mathematik, 135:2, p. 81-133. The first mention of Potenzmenge and the fraktur U notation is on p. 83. Commented Aug 6 at 18:35