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Rating: 
Summary: Insightful
Review: An exceptional book. The book, however, has little pedagogical value. I would not recommend those starting out in mathematics to purchase it. It is definitely for the mathematically mature. Indeed, it is the type of book that may force some to consider abandoning mathematics if it is read without assistance too early in their development. The lack of answers to exercises amplifies these considerations when the book is used for self study as there are few means to understand whether one is on the right track, especially when the less natural approach of recursion is required to answer some questions. If you want to maximise your understanding of set theory, however, this is an essential book to get. In a sense, when read in conjunction with Paul Halmos' background and some quotes attributed to him found elsewhere on the Internet, the book is almost autobiographical.
Rating:
Summary: Insightful
Review: An exceptional book. The book, however, has little pedagogical value. I would not recommend those starting out in mathematics to purchase it. It is definitely for the mathematically mature. Indeed, it is the type of book that may force some to consider abandoning mathematics if it is read without assistance too early in their development. The lack of answers to exercises amplifies these considerations when the book is used for self study as there are few means to understand whether one is on the right track, especially when the less natural approach of recursion is required to answer some questions. If you want to maximise your understanding of set theory, however, this is an essential book to get. In a sense, when read in conjunction with Paul Halmos' background and some quotes attributed to him found elsewhere on the Internet, the book is almost autobiographical.
Rating:
Summary: Mercy Sought
Review: My previous review of Halmos' book stands. Exceptional book, but ... As an example of a question in the book to whet some appetites and in seeking someone's kind mercy in actually answering it for me and putting me out of my misery, consider p.59 on the Axiom of Choice. Quote: if {X (sub)i} is a finite sequence of sets, for i in n say, then a necessary and sufficient condition that their Cartesian product be empty is that at least one of them be empty ... (The case n=0 leads to a slippery argument about the empty function; the uninterested reader may start his induction at 1 instead of 0). Unquote. Induction from 1 is no problem. The slippery argument stuff (and other similar pearls thoughout the book) sends me wild. What is the slippery argument. Please. Anyone. With thanks to Paul Halmos for making my life 'miserably interesting' (sic)!!
Rating:
Summary: Mercy Sought
Review: My previous review of Halmos' book stands. Exceptional book, but ... As an example of a question in the book to whet some appetites and in seeking someone's kind mercy in actually answering it for me and putting me out of my misery, consider p.59 on the Axiom of Choice. Quote: if {X (sub)i} is a finite sequence of sets, for i in n say, then a necessary and sufficient condition that their Cartesian product be empty is that at least one of them be empty ... (The case n=0 leads to a slippery argument about the empty function; the uninterested reader may start his induction at 1 instead of 0). Unquote. Induction from 1 is no problem. The slippery argument stuff (and other similar pearls thoughout the book) sends me wild. What is the slippery argument. Please. Anyone. With thanks to Paul Halmos for making my life 'miserably interesting' (sic)!!
Rating: 
Summary: Mathematical writing at its best
Review: Oh, to be able to write like Paul Halmos! This is, quite simply, a beautiful book. Halmos has taken a field, wrapped his deep understanding around it, and brought the field forth into light in a way that it is accessible to any reader willing to invest the requisite effort, regardless of mathematical background. Each word is carefully chosen; Halmos has a knack for qualifying his statements gently and subtly so that on a first reading, the qualifications and limitations placed on the main results don't slow one down. On a second reading, the qualifications actually shed light on the intricacies of the subject. "Why does he qualify this?", one asks oneself, and in discovering the answer, comes to a better understanding of the field. Similarly, the small number of exercises posed for the reader have been very carefully chosen to she light on the subject itself. Unlike the rote busywork included with many mathematics texts, each problem posed by Halmos is, I would argue, essential to the book. The book is not easy going in that it can be read quickly. I have a reasonable mathematical background, I use mathematics daily in my professional life, and yet (taking time to work the exercises) I read this book at a pace of about four to six pages an hour. On the other hand, this is not so bad - the entire book is only 102 pages, and in those 102 pages Halmos manages to present a full semester's course in set theory. Finally, I should mention that anyone who has spent more time with applied mathematics than with the foundations of mathematics is likely to find this a fascinating read. When I read this book, it was not only the most interesting mathematics book I had read in at least a year, but also the most interesting philosophy book. Just to give a few examples, I never REALLY understood Russell's paradox until I read Halmos' explanation (which he presents on page 6 of the book). By page 30, Halmos offers an explanation of what a function really is, and by page 42, he tackles the question of what we really mean when we talk about the number "2" or the number "6" or any other number, for that matter. This book takes some work on the part of the reader, but the effort is repaid handsomely. The effort would have been worth my while purely to the learn the mathematics, purely for the philosophical issues raised, or purely as an example of how one can aspire to write about mathematics. Of course, for my effort, I was able to enjoy all three aspects of this marvellous text.
Rating:
Summary: The essential essence of set theory in 100 pages
Review: This book is an excellent primer on the basics of set theory that all graduate students need, but are not necessarily obtained in the general undergraduate curriculum. Halmos writes in an abbreviated, yet effective style that imparts the necessary details without an excess of words. Theorems and exercises are very few, so it really cannot be used as a textbook. If you need a great deal of explanations, then it is not for you. However, if your need is for a book that distills the essence of set theory down to the shortest possible size, then this book should be yours, either in your college or personal library.
Rating:
Summary: Very thorough yet too compact
Review: This book is very clear. The style is informal but the details of the rigor are transparent, which is good for every student of mathematics to see at some time. This is especially important because set theory is something that is often used at the foundation of other mathematical works. I'm very pleased that a foundations book can be so accessible to undergraduates.
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