19

Recently I have started reading Bourbaki's Theory of Sets on my own. Regarding one of the explanations of a concept when I went to a Professor of our college, he asked me why I was wasting my time reading a book which contains so many pitfalls. Besides, he also told me that Bourbaki's treatment of Set Theory is wrong. When I asked him about examples, he told me that he thought I would be able to find it myself.

Now this is something that is highly contradicting because here in D. Miller's answer I found that,

...Bourbaki is very far from being suitable for everyone. That said, if you're looking for a reference that's axiomatic, super-abstract, and works in greatest possible generality, and is also crystal clear and careful, Bourbaki is the place to go. ...

And this contradicts the expression I got from my professor. Besides, I himself wasn't able to find out any 'pitfall' in the book so far.

Is there really anything wrong with Bourbaki's Set Theory?

Update

Though Asaf Karagila was very patient with me and answered all my queries, I will be very glad if someone who has gone through Bourbaki's Theory of Sets answers it in detail.

Jyrki Lahtonen
  • 133,153
  • 3
    I find anything by Bourbaki very rigorous, almost to a fault. I know a lot of mathematicians who are put off by the Bourbaki method. However, before Bourbaki mathematics could be said to be in a crisis. That said, I think if this is your first endeavor into set theory, maybe a more modern account would be more appropriate. – Rachmaninoff Nov 23 '14 at 12:02
  • @Rachmaninoff: Actually I was looking for an axiomatic, super abstract book on set theory, and after getting suggestions from this site and consulting with some of my friends, I decided to read Bourbaki. My friends said that the level of rigor in Bourbaki is still unparalleled. Can you suggest me some modern books which meets my criteria? –  Nov 23 '14 at 12:08
  • 11
    I suppose he doesn't mean "wrong" in the sense of "contains false statements as proclaimed 'theorems'" or "contains invalid proofs", but rather "is not the right book to learn se ttheory from, just as Russel-Whitehead's Principia Mathematica are not the right place to learn $2+2=4$ from" – Hagen von Eitzen Nov 23 '14 at 12:10
  • 9
    Note that none of the members of Bourbaki was a set theorist. And that the developments in axiomatic set theory since the 1960's make pretty much any book about axiomatic set theory written before 1970 obsolete. If you are looking for a rigorous approach, pick up Kunen's 1983 "Set Theory", or Halbeisen's "Combinatorial Set Theory" (which also has a free edition on the author's website). Those do a very good job in presenting modern axiomatic set theory. – Asaf Karagila Nov 23 '14 at 12:41
  • I learned it from Pinter's text, which is available in reprint from dover http://www.amazon.com/Book-Theory-Dover-Books-Mathematics/dp/0486497089 . It covers von Neumann Godel Bernay's set theory, so it is worth a look, since others usually take up ZFC. I think if you studied pinter along with Lawvere's Sets for Mathematics, which approaches sets from a category theoretic angle, you would be better off in a year than delving into Bourbaki (but thats just me, you might be able to zip through it like a pro) – Rachmaninoff Nov 23 '14 at 12:48
  • 5
    Maybe this might be an interesting read? – Dejan Govc Nov 23 '14 at 12:49
  • @AsafKaragila: What does 'obsolete' mean in your comment? –  Nov 23 '14 at 13:07
  • 5
    Old and outdated. – Asaf Karagila Nov 23 '14 at 13:33
  • @AsafKaragila: That seems circular to me. Let me make it precise. Do you want to mean that it is wrong or it is simply inconsistent with the modern mathematics but not because it is wrong or something, but because there are simpler and better approaches to Set Theory? –  Nov 23 '14 at 14:19
  • 3
    I didn't write anything about it being wrong or not. I didn't read the book past the first couple of pages that seemed overly complicated to me (even if you want to be strictly formal about set theory). You explained that you chose Bourbaki as a candidate because you wanted an axiomatic "super abstract" book about set theory. I'm not sure what "super abstract" would be. But regardless to that, modern axiomatic set theory has changed its characteristics since the 1960s making the book obsolete regardless to any mistakes. – Asaf Karagila Nov 23 '14 at 14:27
  • @AsafKaragila: Thanks for your patience in answering all of my queries. It helped a lot. –  Nov 23 '14 at 15:08
  • 2
    @MikhailKatz I do not think creating (bourbaki) tag is a reasonable thing to do, considering that result of previous discussion about this tag was strongly in favor of removal. At the very least, discussing the tag on meta first seems like a reasonable thing to do. (As far as I can tell, you are aware of the previous discussion.) – Martin Sleziak Feb 15 '18 at 15:55
  • One example of some "pitfall" in a way is the following: Bourbaki has in its presentation of logic an operator denoted $\iota$. This operator takes a sentence with one free variable $x$ and returns an $x$ such that $\phi(x)$ if such an $x$ exists, and nonsense otherwise. With this operator and standard axioms of ZF, one can easily prove the axiom of choice: this axiom is in some sense embedded in Bourbaki's logic, whereas (it seems to me) set theorists would rather be happy to look at universes where AC doesn't hold. This is an example of a "shortcoming". Surely there are others – Maxime Ramzi Feb 15 '18 at 19:40
  • 2
    @MikhailKatz We have a tradition of clearing suggestions for new tags in meta BEFORE using them. You should argue your case there. That is how we resolve things here. – Jyrki Lahtonen Feb 16 '18 at 06:21
  • @MartinSleziak, I consulted the discussion you linked and found a valid objection by user quid who pointed out that one of the questions included under the tag was a technical algebra question not related specifically to Bourbaki. I agree with quid that the inclusion of that question under the tag "bourbaki" is inappropriate. On the other hand, there is a recognizable bourbaki approach to the philosophy of foundations, and a number of questions here are specifically on that topic. It is actually difficult to find those questions by searching "bourbaki" precisely for the reason user quid... – Mikhail Katz Feb 18 '18 at 17:02
  • ...pointed out, namely, that many questions mentioning "bourbaki" are technical mathematical questions unrelated to bourbaki foundational views specifically. That's why such a tag would be useful. Note that there is an MO tag for bourbaki: https://mathoverflow.net/questions/tagged/bourbaki, somewhat along the lines I am suggesting. – Mikhail Katz Feb 18 '18 at 17:03
  • 2
    @MikhailKatz Clearly (based on your comments and also your attempt to create the tag) you are in favor of (bourbaki) tag, why not making a post on meta about this - as suggested by Jyrki Lahtonen. Your arguments why the tag might be useful would be seen by more users there than in this comment thread. In any case, since the several comments hare are related more to creating tags than to the actual question, I would suggest to continue this discussion in chat. (Or on meta, if you make a post about the new tag.) – Martin Sleziak Feb 18 '18 at 18:56
  • Related: https://math.stackexchange.com/a/1731902/763598 – AKP2002 Dec 28 '21 at 06:17
  • 1
    One thing that's wrong with Bourbaki's set theory is that no one uses it. In that sense, using that book to be your introduction to set theory is a mistake and, ultimately, a waste of time. – Mariano Suárez-Álvarez Feb 12 '23 at 01:07

1 Answers1

1

The fundamental problem with Bourbaki set theory is that they bury the axiom of choice at the level of the syntax of the formulas and not at the semantic level (as an axiom), making the development of different set theories (ZF with or without AC,...) impossible within the frame they chose. A really symptomatic example is the absence of category theory in their treatise, because they would need the Grotendieck-Tarski set theory (ZF + an axiom on universe) to be fully rigorous.

A rural reader
  • 2,507
Sylvain
  • 62