# Which symbol should be used for an empty set?

Currently, a discussion started on the German Wikipedia article for Empty Set (the German discussion), whether $\emptyset$ or $\varnothing$ should be used or is more common as a symbol for an empty set.

Do you know any sources for this question? Do you know a source for the statement, that Nicolas Bourbaki was the first who used this symbol?

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Well, I'd say the macro name \emptyset should be a dead giveaway. BTW, is it intentional that you linked to the English Wikipedia article, although you speak about the German one? – celtschk Aug 21 '12 at 8:32
I think that $\emptyset$ vs. $\varnothing$ is a typographical preference. I have seen both used in set theory books and papers. I am inclined towards $\varnothing$ personally. This is similar to $\mathbb R$ vs. $\mathbf R$ for the real numbers. – Asaf Karagila Aug 21 '12 at 8:57
BTW, the Unicode symbol for the empty set, U+2205, looks like this: ∅. While looking different here, it indeed seems to be what MathJax produces for \emptyset. Whether it looks round or not depends on the used font. MathJax uses a font which resembles what $\rm\LaTeX$ produces. So if this doesn't look round, complain at Donald E. Knuth for designing the font that way. – celtschk Aug 21 '12 at 9:04
No, I have not confused Unicode with a font. I even explicitly said that how that Unicode character looks depends on the font. However the Unicode characters have assigned a meaning, and U+2205 has assigned the meaning EMPTY SET. And I also noted that MathJax correctly generates exactly this Unicode character from \emptyset and that it uses a font which makes it appear in the same style as in LaTeX. And that if you don't like how it looks like, you should complain to the font designer (which in the case of (La)TeX is Knuth). – celtschk Aug 21 '12 at 13:39
My five cents would always be to use $\{\}$... I really don't see how any other, single symbol could be better than this. – leftaroundabout Aug 21 '12 at 18:43

I would say the following should be not too controversial:

• $\emptyset$ and $\varnothing$ are typographical variants of the same mathematical symbol designating the empty set

• The symbol was introduced by Bourbaki, was inspired by the Norwegian character Ø, but is a distinct character from it

• The intention was most probably to create a symbol related to $0$ (zero), not to O (Oh), distinguished from it by striking it through. After all the empty set has all kinds of relations with the number $0$, but none with the letter O. (By contrast big-Oh and little-o symbols derive from the word "order".)

• The symbol has absolutely no relation (apart from appearance) with the lower-case Greek letter phi, with typographical variants $\phi$ and $\varphi$.

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Excellent post, Marc! – Georges Elencwajg Aug 21 '12 at 9:51

There is a webpage where you can find mention of this: http://jeff560.tripod.com/set.html

Here's an extract

The null set symbol (Ø) first appeared in N. Bourbaki Éléments de mathématique Fasc.1: Les structures fondamentales de l'analyse; Liv.1: Theorie de ensembles. (Fascicule de resultants) (1939): "certaines propriétés... ne sont vraies pour aucun élément de E... la partie qu’elles définissent est appelée la partie vide de E, et designée par la notation Ø." (p. 4.)

André Weil (1906-1998) says in his autobiography that he was responsible for the symbol:

Wisely, we had decided to publish an installment establishing the system of notation for set theory, rather than wait for the detailed treatment that was to follow: it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval. Much later, my own part in these discussions earned me the respect of my daughter Nicolette, when she learned the symbol Ø for the empty set at school and I told her that I had been personally responsible for its adoption. The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar.

The citation above is from page 114 of André Weil's The Apprenticeship of a Mathematician, Birkhaeuser Verlag, Basel-Boston-Berlin, 1992. Translated from the French by Jennifer Gage. The citation was provided by Julio González Cabillón.

You should also note that Nicolas Bourbaki is a collective pseudonym for a group of French-speaking mathematicians.

Concerning your question, I remember one of my math professor telling us that in fact, it was a claim by both André Weil and Claude Chevalley, but I can't find the citation for Chevalley.

I've found also this lecture about history of math from a Canadian University (in French), where this issue is mentioned: http://www.mat.ulaval.ca/fileadmin/Cours/MAT-2500/Bourbaki.pdf

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BTW, note that this web page doesn't use the Unicode character U+2205 EMPTY SET (∅), but the Unicode character U+00D8 LATIN CAPITAL LETTER O WITH STROKE (Ø). Ironically, that one looks less round in the input box and in the comment font. – celtschk Aug 21 '12 at 9:12
Excellent post, $S_3MP$ ! – Georges Elencwajg Aug 21 '12 at 9:53
Browsing the Archives Bourbaki is interesting: In the early drafts (before état 2) of Théorie des Ensembles* the symbol $\bigcirc$ was used, e.g. here, page 24 of the typoscript: screenshot. The first occurrence of the $\emptyset$ symbol I found is here, page 46 of the typoscript: screenshot. (Warning: the linked pdf's are huge) – t.b. Aug 21 '12 at 16:07
Dear @t.b., I can't begin to say how admirative of your comment I am. Your erudition is (as usual) truly amazing and I say that as someone who has often browsed the Archives Bourbaki without noticing the facts you report. – Georges Elencwajg Aug 22 '12 at 8:54
@t.b.: Thank you very much for this document! For all others: With huge, he means 108.6 MB! Huge, indeed. – Martin Thoma Aug 22 '12 at 9:55

## protected by Qiaochu YuanAug 22 '12 at 18:55

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