A solid foundation in set theory is one of the essential preconditions of being a good, working mathematician. I have taught nearly every course in the undergraduate mathematics curriculum and the basic concepts of set theory crop up on a regular basis. I have also taught nearly every course in the undergraduate computer science curriculum and those same ideas also pop up on a regular basis. Just last week when I was using finite automata to describe language recognizers it was necessary to remind the class about some basic concepts of set theory.

This book extends the idea of basic set theory beyond what most people would have as a definition. Most of the books in basic set theory place too much emphasis on the basic and not enough on the more advanced topics that students need for future study. Levy thoroughly covers the advanced topics, providing an overview of all of the areas of set theory that graduate students in mathematics need. Point set topology, real spaces, Boolean algebras, and infinite combinatorics are all covered in the second, more advanced part of the book. The coverage throughout is complete albeit succinct. It consists largely of a series of proposition - proof statements with few, if any examples.

This is not a book that a beginner can use to learn set theory, it requires a high level of mathematical maturity before it can be understood. Unfortunately, there are no exercises collected at the end of the chapters, so I hesitate to recommend it for advanced undergraduate or graduate courses in the field. In coverage, it is acceptable, but my personal belief is that all math textbooks should contain exercises at the end of the chapters. It can give you a solid, advanced foundation in set theory, but you have to work hard, slowly and be able to practice on your own without additional problems to work through. It is very well suited for a role as an advanced reference work.

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