I would subtitle this question "Bourbaki's dream." The dream faltered on the foundations. Bourbaki tried to give a half-baked half-formalized naive set theory resulting in an embarrassment of epic proportions that has been detailed by Adrian Mathias, an expert in the field unlike any of the Bourbaki, in a series of recent detailed critiques (not merely his essay The ignorance of Bourbaki). What I am trying to suggest is that checking all of the previous results will get you hopelessly bogged down in the foundations.
Mikhail Katz
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