Elements of Advanced Mathematics

"Preface to the Third Edition On the whole, we have retained the content and character of the first two editions. But we have added material on point-set topology (Chapter 8), on theoretical computer science (Chapter 9), on the P/NP problem (Chapter 10), and on zero-knowledge proofs and RSA encryption (Chapter 12). The topology chapter of course builds on the existing material on real analysis. The computer science chapters show connections of basic set theory and logic with current hot topics in the technology sector. The material on cryptography is exciting, timely, and fun. These new chapters help to make the book more current and significant. It should of course be understood that these four chapters may be considered to be optional. Skipping them will in no way detract from reading the rest of the book. Some readers consider Chapter 5 on axiomatics and rigorous logic to be optional. To be sure, it is a more demanding chapter than some of the others. But it contains important material, some of which is at least alluded to later in the book. Readers who do not want to spend much time on Chapter 5 might wish to at least have a look at it. The main message here is that Chapters 5, 8, 9, 10, and 12 provide an open-ended venue for students to explore and to learn. My experience with teaching this course is that the aggregate material causes many of the students to get really turned on to mathematics. They need to have a means for further exploration and reading. These chapters give them that opportunity, and exercises to back up the reading. The new Chapter 12 is dessert. It presents the very new ideas of zero-knowledge proofs and RSA encryption"--