Product Description

The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. At the same time, it is often viewed as a foundation of mathematics so that in the most prevalent, current mathematical practice "to make a notion precise" simply means "to define it in set theory." This book tries to do justice to both aspects of the subject: it gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets (including the basic results that have applications to computer science), but it also attempts to explain precisely how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author added solutions to the exercises, and rearranged and reworked the text in several places to improve the presentation. The book is aimed at advanced undergraduate or beginning graduate mathematics students and at mathematically minded graduate students of computer science and philosophy.

Product Details

• Amazon Sales Rank: #446936 in Books
• Published on: 2005-12-21
• Original language: English
• Number of items: 1
• Binding: Paperback
• 284 pages

Customer Reviews

Though not a classic (yet!), WAY better than any
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.