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I'm looking for a modern book on logic-set theory-foundation written as the Bourbaki's set theory. I'm particularly interested in a formal exposition of ZFC axiom with logic-set Grothendieck universe.

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  • $\begingroup$ See here: math.stackexchange.com/questions/251490/textbooks-on-set-theory $\endgroup$ – Ryan Sullivant Jul 7 '13 at 23:13
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    $\begingroup$ The two sentences in the question seem to contradict each other. Bourbaki's set theory is rather different from ZFC plus a Grothendieck universe. $\endgroup$ – Andreas Blass Jul 8 '13 at 1:09
  • $\begingroup$ I cite Bourbaki for its exposition, not for the content. Bourbaki starts from the begin and talk about assemblies. I'm looking for a book which introduce a modern foundation for the mathematics starting by describing 'symbols' and 'assemblies'. $\endgroup$ – Fabio Lucchini Jul 8 '13 at 8:32
  • $\begingroup$ There may be some misunderstanding here (though I have to admit I am not familiar with Bourbaki's set theory). The French word "ensemble" means "set" in English. Where do the assemblies come in? $\endgroup$ – Mikhail Katz Jul 11 '13 at 14:09
  • $\begingroup$ Bourbaki's Theory of sets book start from the begin by introducing formal mathematics in which mathematical statmentes are written down by sequences of symbol called assembies. $\endgroup$ – Fabio Lucchini Jul 11 '13 at 16:36
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Bourbaki's book was idiosyncratic, and the terminology it used was never common. Contemporary books do not use terminology such as "assemblies"; they talk instead about "formulas" and "sentences".

The two most common contemporary books on set theory are Set Theory by Kunen and Set Theory (3rd edition) by Jech. Kunen's book is written in the style of a textbook, while Jech's is written in the style of a desk reference, but each book can serve both purposes. Both of these books start at the beginning and present set theory in a fully rigorous manner.

The existence of Grothendieck universes is equivalent to the existence of strongly inaccessible cardinals. Inaccessible cardinals are very well understood in set theory, so set theorists don't spend much time also talking about Grothendieck universes.

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  • $\begingroup$ I'm not sure that the first half of Jech is a desk reference. $\endgroup$ – Asaf Karagila Jul 8 '13 at 11:43
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    $\begingroup$ I read first pages of both books, but the exposition don't seem complete and rigourus as in Bourbaki. For example, Bourbaki gives a precise distinction between logic symbols, letters and specific signs; theire usage in defining what is a term or a relation; he give a precise definition of quantificators and os on. Both Kunen and Jech seems does not give such costructions. $\endgroup$ – Fabio Lucchini Jul 11 '13 at 21:23

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