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$\begingroup$

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?2012-08-21
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    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.2012-08-21
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    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.2012-08-21
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    The design of the \emptyset symbol used in $\rm\TeX$ (and I think unchanged in $\rm\LaTeX$) is perfectly well documented, in volume E of DEK's Computers and Typesetting. I don't have the book at hand right now, but I bet the program drawing it is exacly that of the digit $0$, with a diagonal line added to it. So if this doesn't look round, that is because $0$ doesn't either.2012-08-21
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    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).2012-08-21
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    This is not a constructive question... It's just about "what symbol is better"... it's a matter of preference!2012-08-21
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    Judging by two excellent answers, I think this is not a bad question at all...2012-08-21
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    @Asaf: I'm confused: your inclination seems to depend on whether I [show the hidden comments](http://i.stack.imgur.com/eenWa.png) or [not](http://i.stack.imgur.com/BVLje.png) :)2012-08-21
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    It’s clearly a matter of taste. I’ve a strong enough preference for $\varnothing$ to find $\emptyset$ quite jarring, like a bad compromise between the ordinal $0$ used for the empty set and the special symbol $\varnothing$ for it. I also think that $\varnothing$ is closer to the original inspiration for the symbol.2012-08-21
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    My five cents would always be to use $\{\}$... I really don't see how any other, single symbol could be better than this.2012-08-21
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    @leftaroundabout: By introducing less clutter. There are already more than enough brackets of various sorts in many of the expressions in which $\varnothing$ appears.2012-08-21
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    @leftaroundabout: It is not an uncommon question in introductory set theory courses to ask how many elements are in the set $\{\emptyset\}$. I think this would get lost when asking about the set $\{\{\}\}$...2012-08-22
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    @user1729 Perhaps even more people would initially think "$\bigl|\bigl\{\{\}\bigr\}\bigr| = 0$" than "$\bigl|\{\emptyset\}\bigr| = 0$", but that's not a bad thing: after all, the main point of this question is to get the sometimes counterintuitive properties of the empty set clear. Once they've grasped it, I believe $\bigl|\bigl\{\{\}\bigr\}\bigr| = 1$ is in fact _more_ memorable and clarifies the matter very well. $\bigl|\{\emptyset\}\bigr| = 1$ doesn't look that surprising, not more than $\bigl|\{0\}\bigr| = 1$, though there is quite a big conceptual difference between these two.2012-08-22

2 Answers 2

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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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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.2012-08-21
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    Excellent post, $S_3MP$ !2012-08-21
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    Browsing the [Archives Bourbaki](http://portail.mathdoc.fr/archives-bourbaki/) is interesting: In the early drafts (before état 2) of Théorie des Ensembles* the symbol $\bigcirc$ was used, e.g. [here](http://portail.mathdoc.fr/archives-bourbaki/PDF/053_iecnr_061.pdf), page 24 of the typoscript: [screenshot](http://i.stack.imgur.com/3CMdY.png). The first occurrence of the $\emptyset$ symbol I found is [here](http://portail.mathdoc.fr/archives-bourbaki/PDF/055_iecnr_063.pdf), page 46 of the typoscript: [screenshot](http://i.stack.imgur.com/VxgXj.png). (**Warning:** the linked pdf's are *huge*)2012-08-21
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    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.2012-08-22
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    @t.b.: Thank you very much for this document! For all others: With huge, he means 108.6 MB! Huge, indeed.2012-08-22