This is a superb book, but it has a very specific audience. It is a careful, systematic, investigation of the extent to which the methods of set theory can be used to address philosophical questions. So the audience needs to both be comfortable with the formal presentation of mathematical theories, and to know the issues in the philosophy of mathematics. If you lack the philosophical part, you'll wonder why Potter doesn't just use ZF, and why he keeps being drawn off into various topics along the way. If you lack the mathematical part, you'll find the book hard to understand, although it is extremely systematic. (If you don't know what ZF is, for example, I'd advise starting with some other book.)



Having said that, Potter goes out of his way to present matters clearly and explicitly. Readers who don't exactly fit the audience will learn an enormous amount from this book. Moreover, it is so clear and authoritative, and covers so much ground, that it deserves to be in the canon. It ought to displace Quine's Set Theory and its Logic, for example.



ZU is Potter's set theory (76). It is spare, and very powerful. I believe Potter is trying to capture as much as he can of Frege's original view of sets as logical objects, although he doesn't say this. ZU allows flocks of doves and packs of wolves to be sets, just as it intuitively ought to, but it can also capture the real and transfinite numbers. The book divides into four parts. First, there is the presentation of ZU and its properties. Then we get the usual development of the real numbers. The third section deals with ordinals and cardinals, and a fourth section the axiom of choice and the continuum hypothesis.



What sets the book apart, though, is its constant return to the history of its subject and the philosophical issues that have been embroiled in it up to the present. You can look through the book at any issue that interests you - Russell's paradox, non-standard analysis, whether there is some deeper notion of a collection underlying set-theory - and Potter always gives a clear explanation and has something interesting to say about it. With graduate students of sufficient ability, the book would make for a really worthwhile graduate level course in philosophy.



When I'd finished reading it, I wanted to read it again.