Notes on Set Theory (Undergraduate Texts in Mathematics) [Englisch] [Gebundene Ausgabe]

This book is a descent introduction to Set Theory. Whether you want a textbook for a relevant course, or plan to use it for self-study, this should be your choice.

The overall writing style is rather friendly and a bit humorous at times, never tiring (hardly reading this book, will you ever say "oh, come on, why do you do that, it's boring"), but still rigorous; everything is carefully developed with no gaps nor hidden important details. Some interesting historical remarks and quotations can be found along the way, and exercises help the reader test their knowledge on new concepts. Solutions for them are provided in this, second edition, and Problems accompany the end of each chapter, ranging from easy to some very difficult ones. It's written in the "paragraph" style, in the sence that chapters are not subdivided in sections and then in subsections and then in subsubsections, etc.: each chapter deals with a lot of relevant ideas which progress steadily.

Regarding what the book covers, I urge you to look at it's pretty concise table of contents. In a nutshell, it covers the basic concepts of equinumerosities, countable-uncountable sets, the paradoxes, and then starts the axiomatic approach (chap. 1--3). The author is very careful to prove and develop as much as it is possible to do so with each new axiom that is introduced, sometimes at the cost of longer proofs, but overall it pays when it comes to comprehension of the material. When new machinery is introduced, some then-difficult proofs become now-easy exercises. Constructions for the representation of common mathematical objects like ordered pairs and functions follow, a weak, but possible at the moment cardinal assignment and the relevant arithmetic is introduced (chap. 4) and then the natural numbers are also constructed (chap 5.). Then (chap. 6--7) partially and well ordered sets are discussed, fixed points, partial functions, graphs, streams and some concepts from topology are introduced. Up to this point, the reader patiently works without the Axiom of Choice which comes into play at chap. 8, and even then, whenever possible ---which actually is, most of the times--- the weaker version (Dependent Choices) is used. With the machinery of AC the book continues to some of it's consequences, and then proceeds to discuss Baire space, analytic (Suslin) and perfect pointsets (chap. 10) where also some important theorems from logic are stated, quenching the curious and impatient reader. The axiom of Replacement (chap. 11) is then introduced and the relevant axiomatic theories discussed, and so are some other axioms, like the principles of Purity and Foundation. Ordinal arithmetic and the definition of cardinals (von Neumann's, Frege's and Scott's) follow (chap. 12), together with some problems dealing with important ideas like strongly inaccessible cardinals. The two appendices cover a "somewhat novel" construction of the real numbers, which lies between Cantor's and Dedekind's, and for the more "logically" minded, the second appendix is devoted on set universes, including Rieger universes and Aczel's Antifounded Universe.

A lot of recursion/induction is presented and used throughout the book, each time on a different background: complete, simultaneous, with or without parameters, on natural numbers, on ordinal numbers, transfinite and so on.

On the book's level: Well, the author makes everything absolutely clear, (something hard to accomplish on such a field), and also makes everything seem very interesting (something easy to accomplish on such a field). That being said, I think that a mature reader who studies the book and attacks the problems, will have no problem following most of it. It delves a bit deep in some difficult subjects, but one is free to omit those on a first reading, without losing the general concept.

Comparing the second edition with the first, definately the second one is better. A lot of improvements were made, some annoying and maybe confusing typos were corrected, solutions to all exercises in the text were added and generally, there isn't anything valuable in the first edition, missing in the second one. A lot of proofs are clearer, new interesting problems can be found at the end of the chapters and generally changes have been made towards a better understanding of the material.

Finally, i'd like to add that I've also studied Enderton's and Suppes' books, and read through most of the relevant books I could find and this one is clearly the winner: More enjoyable, more interesting, clearer, and better structured. It also "dares" to touch more subjects, poking the reader, motivating for deeper study of those areas.

Reading my review I can only think that I failed to capture just how good this book is. This (along with Hardy's Pure Mathematics) is the best book I've ever studied. Really.

I took two courses from Professor Moschovakis which used this book as their text. I highly recommend this book for anyone who has always wanted to learn about set theory. It starts off with Cantor's naive concept on set theory and his proofs concerning the "sizes" of infinite sets. From there, it presents Russell's Paradox, which created the necessity for the axiomatic system that Zermelo developed in 1908. From there, the book discusses the "construction" of the natural numbers, cardinal/ordinal arithmetic, partially and well-ordered sets, the results of the Axioms of Choice and Replacement, and much more. The only thing that I would criticize this book for is that sometimes the theorems that you'd really like to see proofs of are left as homework assignments. Other than that, this book rocks!

To understand this book you need to know the authors's background, since he covers material most relevant to it, and not necessarialy that of the general subject of set theory. Moschovakis is a died- in- the- wool recursionist who stuudied under Kleene, as did his wife Joan. He is most know in the math world as an expert on Descriptive Set Theory, which is set theory of Polish Spaces and a very speciialized topic. So if you want to get your set theory from this specialized point of view, read his book, otherwise forget it.

I have studied the book thoroughly and can tell you it is the work of a master recursionist. The problem is that although set theory crosses boundaries with recursion theory, it is not althgether driven by it as this book mostly implies. Witness the chapter on fixed points and the way the next chapter on well ordering is handled. Also the emphasis on the dependent axiom of choice. Then the authors' favorite topic- Baire Spaces. This man is trying to make you into a clone of himself--a Descriptive Set Theoryist! On the other hand, the treatment of cardinal arithmetic is exceptionally light, no mention of Hausdorffs' formula, not to mention Silver's theorem or higher cardinals.

Regarding how the book is written-it gets an A for interesting, but an F for buryiny the reader under recursion. (By the way, recursionist are people who talk about something strangely by only talking about something before it or after it, but never it itself.) I will not buy the second edition since I have the first well worn, but I suffered through numerous typos and errors, which hopefully you won't.

Unless you are an aspiring, Descriptive Set Theorist(eg, in love with Polish Spaces), you will be much better served by more traditional texts such as Levys' Basic Set Theory, which is written on a similar advanced level.