Customer Reviews

Classic Set Theory for Guided Independent Study (Chapman & Hall Mathematics S.)

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I went to my university's library and tried to compare Goldrei's book with almost all other popular accounts on Set theory, prior to investing my time learning the basics of the subject. After having almost finished reading it, I believe that this a model of how textbooks in mathematics should be written. Let me be a little bit more specific:

1)Suberb organization of the material. Ideas are built gradually without logical gaps or regressions. Definitions and theorems are clearly stated. "What follows from what" is always transparent. Even the choice of paragraphs is so well thought, that one can easilly assign a title in each of them for quick reference later on.

2)Each subject is clearly introduced within its historical and logical context. Each theorem (and even exercise) is motivated for its importance and its merits in the global picture of Set theory.

3)The logic and intuition behind the proofs is given (as well as the proof itself...) in a well organized and not unecessarily wordy manner.

4) There are exercises within the main text (which, as usual, are well motivated for their importance) with solutions folowing right after. In this way, one may develop skills and understanding, without getting frustrated or spending too much time. There are also exercises in the end of each section which are interesting and not too difficult.

5) There are comments aside of the main text, which range from ideas concerning a proof to historical remarks or recommendations to the reader. In this way, the main text remains clean of tangencies, but never dry.

I could continue praising this book, but let me cut it short by saying just this: it is one of those proper (i.e. rigorous) math textbooks that invite you to read each following chapter and to turn each page to see what's next. Having finished it I feel I have a pretty firm understatnding of the basics.

I only wish that Goldrei could write a second book on specialized topics (say, similar to the topics covered in Devlin's "The Joy of Sets", or Moschovakis' book), with the same energy and enthousiasm that wrote this one.

In recent months, I read (at least in part) about ten books on set theory, of different levels and styles. Goldrei's book is by far the best on my list. Goldrei has the talent of presenting a fairly difficult and abstract material in a friendly style, with abundant explanations, solved exercises, illustrations, and illuminating comments. I can highly recommend this book as a first reading on set theory, whether for self-study or as a supplementary text for a course on the subject. For the interested readers of this review, here is how I rank some of the other books I've read: Stoll is definitely a second, then Suppes (which goes deeper than both Goldrei and Stoll, but in my opinion is poorly organized and suffers from nonstandard and funny-looking notations), followed by Halmos, Machover, and a few more. And last and least, the incomprehensible "book" by Vaught.

What this book tries to do, is to make set theory understandable for everyone. The book starts with the construction of the real and natural numbers. Proceeds with Zermelo-Fraenkel set-theory, and goes on to discuss the axiom of choice, cardinals and ordinals, and finally the continuum hypothesis.

Considering the subject matter, the book is extremely easy to read, with an easy progression in just under 300 pages. It never looses focus to get bogged down in detail, but somehow still manages to take you through all the essential proofs. The only thing missing is a few more interesting exercises for those wanting to refine their proof skills.

Even if you have only a passing interest in logic (or the foundations of mathematics), this book is an excellent starter.

Recently, I had to learn set theory for my current work (automated reasoning and expert systems). This book proved to be an excellent supplement for learning set theory; which I then used to learn formal logic, resolution, etc.

Very well written and with good examples and exercises.

This book is especially useful for self study, as it has more worked examples than usual, motivating hints in side bar, and explanitory comments in text. Perhaps the origin of this materical in Open University (I think roughly analogous to what USA schools call "Extension"?) course is reason. Found it helpful where Halmos was a bit concise for someone with fellow students or TA handy.

With the exception of the very final section and a few bumps in chapter five, this was a very smooth read overall. Proofs were very clear and thorough, and the reminders of previous definitions in the margins were extremely helpful, so I didn't have to turn back the pages. No real prerequisites are needed (analysis is recommended, but I wouldn't worry too much about it); my background in mathematics is relatively weak and I got through it pretty easily. My main complaint was that he started by working backwards, breaking down the reals & then the naturals before getting to the axioms themselves. But this is a personal preference, and should not be a big deal to most readers. I also found that it was a bit too exciting, as all the exclamation points (with hardly a mention of factorials) interrupt the flow of the book.

This book should be the first on your list if you plan to study set theory or foundations of mathematics in general. The writing style is informal but the material is well presented to the reader in a way that its rigor is not sacrificed. The author goes through many proofs outlining each and every step. Solutions to exercises are also included even though as the book progresses the author omits more and more solutions because he expects that the readers level should be increasing. I really liked this method. I have not read other books on set theory, but it is really hard to imagine a better one. Not much is expected as prerequisites. If you understand the mathematical concepts of convergence, sequences and functions then the book should be pretty much self-contained. If not I would recommend that you read something like "Sets, sequences & mappings" by Anderson and Hall. This is a really small book but its excellent in that it prepares you for more abstract courses.