Adrian Mathias has written a number of excellent essays criticising various aspects of Bourbaki's logical foundations, and I encourage you to follow the link and read them. He writes supremely well, and some of these essays are simply riotous, when he exposes particularly ridiculous aspects of the Bourbaki systems, such as the last item below.
Probably the main reference you should look at is:
- Adrian Mathias, The Ignorance of Bourbaki (a commentary on the foundational stance of the Bourbaki group. In: Mathematical Intelligencer 14 (1992) 4--13 MR 94a:03004b, and also in Physis Riv. Internaz. Storia Sci (N.S.) 28 (1991) 887--904, MR 94a:03004a. A translation by Andras Racz into Hungarian is available, under the title Bourbaki tevutjai, in A Termeszet Vilaga, 1998, III. kulonszama.)
In this essay, he is highly critical of the Bourbaki stance on set theoretic foundations, and seems to view it as taking place in a bizarre historical vacuum. Although they discuss various historical contributions to set theory, they do not mention Goedel and his towering contributions. Their set theoretic system, amounting essentially to the Zermelo axioms with choice, is strangely weak, insufficient for many mathematical constructions. (For example, in ZC, you cannot prove that the ordinal ω+ω exists, or that there are any sets of cardinality Alephω.)
Mathias has several follow up articles on his web page, continuing the discussion of this topic, and his essays now form a dialogue with various writers defending Bourbaki. For example, he has interesting articles engaging with Mac Lane and with Mac Lane's set theory, which shares similarities to that of Bourbaki.
Finally, there is his charming essay lampooning the Bourbaki formal system, while also giving a thorough logical analysis of it
- Adrian Mathias, A Term of Length 4,523,659,424,929, Synthese 133 (2002) 75--86.
He describes it thus:
A calculation of the number of symbols required to give Bourbaki's definition of the number 1; to which must be added 1,179,618,517,981 disambiguatory links. The implications for Bourbaki's philosophical claims and the mental health of their readers are discussed.
(I mentioned these essays also in this related question.)
