Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the discussion, the author asserts that the Bourbaki group never acknowledged Godel's results on incompleteness or Russel's paradox.

However, from the introduction to their Theory of Sets, one can read:

To escape this dilemna, the consistency of a formalized language would have to be "proved" by arguments which could be formalized in a language less rich an consequently more worthy of confidence; but a famous theorem of metamathematics, due to Godel, asserts that this is impossible for a language of the type we shall describe, which is rich enough in axioms to allow the formation of the results of classical arithmetic.

And further,

Indeed, this is more or less what has happened in recent times, when the "paradoxes" of the Theory of Sets were eliminated by adopting a formalized language essentially equivalent to that which we shall describe here; and a similar revision would have to be undertaken if this language in its turn should prove to be contradictory.

So, is Theory of Sets by Nicolas Bourbaki as outdated and obsolete as A. R. D. Mathias suggests? If so, how does that affect the subsequent volumes, if at all?

Ideally I would want to read several of them by starting, for completeness and coherence's sake, with vol. 1, having already a solid grasp of basic set theory (at the level of Hrbacek & Jech), after making the connection with mathematical logic (at the level of Enderton) and derive other theories from there (in the spirit of the Bourbaki's), for a personal write up.

Edit: For those stumbling on this, Mathias does indeed seem to overlook several elements indicating that Bourbaki were very well aware of pretty much everything he calls them out for. I recommend reading all the historical notes interspersed within Bourbaki's Theory of Sets, especially the very last one which is quite informative.

Solution 1:

You may want to read this critical review by S.L. Segal of that essay:

https://zbmath.org/?q=an:00096766

As Segal points out, Mathias is mostly raging against the neglect of set theory, logic, and foundations as worthy subjects of study. In a further response to Segal's review Mathias admits that in his essay he was not attempting to be a 'sober historian'.

Altogether that essay is more or less a personal rant, not serious academic output. It is an invective against Bourbaki-influenced mathematicians for not taking logic seriously. It blames Bourbaki for the dismissive attitude towards mathematical logic and foundations that exists in the mathematical community. Mathias laments that Bourbaki did not deem Gödel's work as worthy of being included in a volume on set theory. This is what he means by Bourbaki's neglect of Gödel, not that Bourbaki's Set Theory is inconsistent, but that it's not an in-depth treatise on mathematical logic.

For academic purposes you can safely ignore any mathematical concerns in that essay and not be any worse off for it. You can likely find much more serious critiques of Bourbaki addressing similar concerns.

Bourbaki's treatment of set theory and foundational material is outdated. It's only meant to provide a solid starting point for the 'real math' in the subsequent volumes, not to study set theory in itself. For its own purpose it is entirely adequate.

One of the shortcomings of Bourbaki's Set Theory as foundations for math is that there is no mention of categories anywhere. Instead it uses rather contrived constructions such as 'structures' and 'species'. This language is now almost entirely extinct. Additionally, for ideas such as 'adjoint functors' there are no alternative constructions offered at all, and they are entirely absent throughout the volumes.

Don't read Bourbaki's Set Theory if you want to understand set theory as a mathematical field. Personal interest or wanting to understand Bourbaki's at-times arcane language are better reasons to look in that book. In any case it's not a very difficult read, but it contains extremely convoluted constructions for basic mathematical objects, and many of these constructions are meant to be forgotten once their existence is confirmed.

In fact for learning math it's probably a good idea not to read too much Bourbaki. Reading a few passages here and there is likely beneficial, but reading the entire series from beginning to end is probably a waste of time, as each of the general topics covered have advanced since and have their own modern canonical textbooks.

Solution 2:

Mathias does not merely criticize a neglect of logic and set theory, but also documents serious errors in Bourbaki's Theory of sets. There are both conceptual errors, when an outdated formalism involving Hilbert's $\tau$ is used throughout, and also technical errors when theorems are incorrect due to missing hypotheses. There is further confusion between language and meta-language. None of those involved in writing the volume and its sequels by Godement and others are professional logicians.

Instead Weil (and Bourbaki following him) relied on anecdotal mail exchanges with Rosset who apparently didn't pay too much serious attention to this. The $\tau$ assumption turns out to be stronger than the usual form of the axiom of choice customarily assumed in ZFC. As Mathias puts it, it is somewhere between the usual AC and global choice. I find Mathias' conclusions concerning teaching logic in French highschools not that well supported, but his mathematical analysis of the mistakes in Theory of Sets and its sequels is spell-binding.

In this sense the Bourbaki treatment of set theory is not merely outdated; it is refuted.

Many of the participants in this discussion have read Mathias' "The ignorance of Bourbaki" but some may not have read his much longer piece "Hilbert, Bourbaki, and the scorning of logic". This can be found here.

Solution 3:

Any argument in any of the chapters following the sections on formal systems would be accepted by a vast majority of modern working mathematicians (many of which would also happily accept the arguments on formal systems as well).

Consequently, if it is your purpose to study and learn the art of rigorous mathematical argument, then you will find Bourbaki to be just as timelessly relevant as Euclid's Elements (perhaps more so for Bourbaki has a number of interesting practice problems at the end of relevant sections).

If it is your purpose to study and use modern tools for formalizing modern mathematical systems, then Bourbaki is unlikely to give you the insight you desire (outside of its use as a tool for learning the art of rigorous mathematical argument, as I've already mentioned).