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Which is the best book on axiomatic set theory? I am interested in a book that is suitable for graduate studies and it is very mathematically rigorous.

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    $\begingroup$ A related question: What is a good text in intermediate set theory?. $\endgroup$ – Martin Sleziak Jan 28 '12 at 6:21
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    $\begingroup$ There is no "best book". $\endgroup$ – Asaf Karagila Jan 28 '12 at 8:53
  • $\begingroup$ @Asaf: Interesting point. I reminded me that, as opposed to John Rawls, Amartya Sen claimed that there is no "fairest society" in the language of elementary set theory in his book the ideas of Justice. $\endgroup$ – Metta World Peace Nov 29 '12 at 1:16
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I've found Kunen's book "Set Theory: an Introduction to Independence Proofs" to be very good. I've heard Jech's book is good also.

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  • $\begingroup$ I came here to post Kunen also. $\endgroup$ – tomcuchta Jan 28 '12 at 6:11
  • $\begingroup$ There is also a new version of Kunen's book which cover more recent topics. $\endgroup$ – azarel Jan 28 '12 at 6:13
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The best textbook on axiomatic set theory for Zermelo-Fraenkel Set Theory is "Axiomati Set Theory" by Patrick Suppes.

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Herbert Enderton's book at an upper-division undergraduate level is not bad.

At a more advanced level, there's that of Azriel Levy, but this one still doesn't get into the metamathematics.

The two above both explain the axiom system.

Halmos' Naive Set Theory is called "naive" apparently because he views sets as collections of objects rather than as whatever-satisfies-the-axioms. Even though it does that rather than explaining ZFC, it may be worth reading.

Kamke's Theory of Sets is also not "axiomatic" but I seem to recall learning some good stuff from it. I think it was from that one that I learned that the indices in the sequence $\aleph_0,\aleph_1,\aleph_2,\ldots$ are just the ordinals, including e.g. $\omega$ and $\omega+1$ and so on, and the least uncountable ordinal, etc. He goes through a very careful derivation of the well-ordering theorem from the axiom of choice.

Another interesting "naive" book is by Naum Vilenkin. I think that may be the one from which I first learned the diagonal argument: the line is uncountable, even though the rationals are countable and a countable union of countable sets is countable.

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I recommend "The Foundations of Mathematics" by Kenneth Kunen for a first approach.

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