# $\aleph$ looks like $\mathbb N$?

We all know the notation $\aleph_\lambda$ for the $\lambda$th (or, I guess, $\lambda+1$st) infinite cardinal number; in particular $\aleph_0$ is the cardinality of the the set of natural numbers $\mathbb N$.

Out of curiosity:

Is it the case that historically, the Hebrew letter $\aleph$ (aleph) was chosen because it sort of looks like the letter N?

• The $\aleph$ notation appears in Cantor's Contributions to the Founding of the Theory of Transfinite Numbers. As far as I can tell, N wasn't used there to denote the set of natural numbers. This suggests that the answer is 'no'. Apr 8 '14 at 22:07
• It's a shame, too. At least on two occasions on math.SE someone used $\aleph$ to denote $\Bbb N$. And not to mention people who are not native Hebrew speakers writing $\aleph$ in all sort of ways which are not even wrong... But then again, it's always nice to have my students recognize a mathematical symbol and acting all surprised that it's not just Greek letters and weird symbols! Apr 8 '14 at 23:38
• @AsafKaragila Have you seen the MathSciNet review of the second edition of Bourbaki's set theory (MR0154814)? The last sentence reads: "In the first edition, all alephs except those appearing in exponents were printed upside down; in the new edition the exception has been removed." Apr 9 '14 at 0:35
• @Andreas: Quite amusing. I suppose that at the time it wasn't trivial to find an $\aleph$ glyph in Europe... Apr 9 '14 at 0:38

According to not necessarily reliable internet sources, Georg Cantor "told his colleagues and friends that he was proud of his choice of the letter aleph to symbolize the transfinite numbers, since aleph was the first letter of the Hebrew alphabet and he saw in the transfinite numbers a new beginning in mathematics: the beginning of the actual infinite."

Edit: According to less sketchy internet sources "The choice was particularly clever, as Cantor was pleased to admit, because the Hebrew aleph was also a symbol for the number one. Since the transfinite cardinal numbers were themselves infinite unities, the aleph could be taken to represent a new beginning for mathematics." from 'Georg Cantor and the battle for transfinite set theory' at http://ad.infinitum.simons-rock.edu/Dauben-Cantor.pdf, with footnote:

"Cantor explained his choice of the alephs to denote the transfinite cardinal numbers in a letter to Felix Klein of April 30, 1895. The original letter is in the Klein Nachlass, Universitatsbibliothek, Gottingen, and may also be read in a draft version in Cantor's letter-book for 1890-1895, pp. 142-143, also kept in the archives of the Niedersachsische Staats- und Universitatsbibliothek, Gottingen. See also Dauben 1979/1990, pp. 179-183; Meschkowski 1991, pp. 354-355."

• The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity By Amir D. Aczel Apr 8 '14 at 22:04
• @BjørnKjos-Hanssen I do not have the book, but here is a quote from a review "the number of whole numbers is aleph-null; the number of irrational numbers, aleph-one." It is not clear to me if this mistake comes from the author or from the reviewer. publishersweekly.com/978-1-56858-105-7 Jun 24 '18 at 12:02

There is another explanation for Cantor's choice of aleph: not the numerical value of the character, but its occurrence in the word (phrase?) denoting infinity. Here is a quotation from the article by Yuval Ne'eman, Issai Schur died here: some background comments, in memoriam (xxi–xxx)"MR1985185 (regarding Jewish mathematics professors): "Another interesting case is that of Georg Cantor (1845-1918), probably the most original and creative mind in nineteenth century mathematics. In this case, conversion to Christianity had already taken place in his parents' generation, but he identified with Jewish destinies and used the Hebrew letter aleph $\aleph$ for systematics of infinity (in Hebrew, ein-sof) which starts with an $\aleph$ and for which he was criticized by editors".

• I highly doubt this explanation, as prior to the birth of modern spoken Hebrew, the Hebrew word for infinity ("ein-sof") only appeared in esoteric mystical texts in Hebrew, and I'd be quite surprised to discover that Cantor (who was actually christian) was familiar with those texts.
– Haim
Apr 9 '14 at 1:31
• Cantor may not have had much contact with Judaism, but I suspect that many of the Protestant theologians with whom he did have contact would have had a good working knowledge of Hebrew: certainly biblical Hebrew, but maybe also some of those esoteric texts. Apr 9 '14 at 4:17
• I tend to agree with @Haim on this issue. This sounds more like a later interpretation that would have earned a very large [citation needed] or some other Wikipedia notice for lack of citations or foundation. Apr 9 '14 at 4:21