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.
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So, is Theory of Sets by Nicolas Bourbaki as outdated and obsolete as A. R. D. Mathias suggests?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.