I am wondering if there exists a book that discuss different axiomatizations of set theory and compare them. Can you please give me any references?
3 Answers
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Foundations of Set Theory by Fraenkel, Bar-Hillel and Levy is a classic that provides what it sounds like you're after. It surveys ZF and its milieu, type-theoretic approaches (including Quine's New Foundations, for instance), intuitionism, and more.
Though I don't know your exact goal, or how specifically your interest is tied to set theory as opposed to, say, somewhat broader foundations of math concerns, you might find Steve Awodey's "From Sets to Types to Categories to Sets" to be useful. It is a short, focused and enlightening look at these prominent approaches to foundations in relation to one another, and it offers a nicely ecumenical account. E.g. this passage from the concluding section:
the objects of type theory and set theory are structured by the operations of their respective systems in certain ways that are not mathematically salient. That additional information is essentially what is lost by our comparisons ... Categorical structure is closer to the mathematical content, and it is not lost in translation. ... The structural approach implemented by category theory is thus more stable, more robust, more invariant than type or set theoretic constructions. On the other hand, type and set theory have certain distinctive advantages as well. ...
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If Ed Dean's suggestion ("Foundations of Set Theory" by Fraenkel, Bar-Hillel and Levy) is to heavy for you, have a look at "Intermediate Set Theory" by Drake and Singh. One chapter describes/discusses several different axiomatizations (Zermelo-Fraenkel, Neumann-Bernays-Gödel, Morse-Kelly, Montague-Scott, Quine, Ackermann).
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And if you happen to be (able to read) Czech try Sochor's Metamatematika Teorii Mnozin