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I'm currently planning on reading Suppes' Axiomatic Set Theory, because I'm interested in finding out what the currently accepted foundations of mathematic are. Is this a good book for doing so? What other good texts are there?

I am not looking for naive set theory, though I do not wish to go more deeply into the subject than the set theory that is needed as foundations of most mathematics today.

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Peter Smith has a very good book list on his "Teach Yourself Logic" page. There is also a nice set of lecture notes by Stephen Simpson located here: "Foundations of Mathematics" (PDF).

One question that you should clarify for your own benefit is whether you are interested in learning about foundations of mathematics, or about set theory. Set theory is an important tool in the field of foundations of mathematics, and it is also a topic of study in its own right. So, for example, well-regarded books such as Kunen's Set Theory: An Introduction to Independence Proofs and Jech's Set Theory will teach you a lot of set theory, but they will not teach your much about foundations of mathematics.

If you are genuinely interested in foundations of mathematics, the literature list is more difficult, because the topics are spread around many fields of math, and there is no single reference that will contain everything, much less present a coherent view of foundations.

If you are interested in the role set theory plays in foundations (compared to the study of set theory for its own sake), one very nice book is Foundations of Set-Theory by Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy. Levy's book Basic Set Theory is actually a graduate level text, is also very good at emphasizing foundational issues, and is now available from Dover.

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  • $\begingroup$ Good answer. One of the difficulties is that many of the people who care about foundations of mathematics will often disagree about what is "the right foundation". So there is no book to encompass basics of logic, set theory, model theory, proof theory, type theory, other set theories, other type theories, categories as foundation, intuitionistic and constructive foundations, and so on and so forth. $\endgroup$ – Asaf Karagila Feb 21 '15 at 14:34
  • $\begingroup$ I suppose I simply meant an introduction to the accepted set of axioms that most mathematics is based on, which to my knowledge is ZFC set theory, though I would welcome a text covering other topic which supplement that. I'm not entirely sure what proof theory or model theory is, but I could look that up. $\endgroup$ – Nethesis Feb 25 '15 at 8:43

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