Naive Set Theory Paperback – August 17, 2011

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.

This is an edited version of my original review. First impressions don't always last! Today I find it horrible as a reference. It's just too wordy. Why not use a few equations instead of making lengthy explanations in words? Even beginning math students are supposed to learn the FORMAL language of math, so why not use it at the outset? The rest you read below is my original review (without change). I didn't change my original rating, but today I'd definately rate it lower!

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There is no escape from Set Theory in mathematics, and by extension, in physics. I finally realized that and went to the basics and bought this book and I am glad I did. Every little piece of knowledge I have in mathematics now appear to me in a brighter light.

The book starts from scratch in that it assumes no prior knowledge in mathematics at all. It does, however, assume knowledge of basic pure logic. Set Theory is developed through the introduction of the axioms, one by one, where the axioms are taken as universal truths which cannot be derived (from previously introduced axioms).

This development goes through various theorems valid for all sets, like De Morgans laws, the formation of new sets from old ones, like the power set and cartesian products, relations a other more specialized constructs, like functions.

Special sets are developed, e.g. the natural numbers. It is an amazing experience the first time one realizes that all sets one need (that I know of) in mathematics can be constructed from the emtpy set. Even more amazing is the fact that most of the symbols used in mathematics are actually sets.

The development goes through ordinal numbers and their arithmetic, and end with a brief introduction to cardinal numbers. Along the way one gets some insight into the precise meaning of infinite numbers and it's a thrill to discover that it's clear that one infinite number can be very much larger than another. In the same context it's also a little amusing to see that one can't push things too far even when one is in the realm of uncountably infinite numbers (quote "...there is no set that big...").

This book clearly deserves five stars, there is no doubt about that. I agree with what most other positive reviews say, but I would like to point out a few shortcomings:

The book could have been clearer; there are in my oppinion sometimes too many scentences and too few equations. In the same way I believe that there are too many words in the equations that are there. Longer statements with the ubiquitous "If and only if" and "for some" and the like become tiresome and even bring linguistic intricasies into the picture. They can and should be replaced by symbols.

Negative numbers aren't even mentioned. Rational numbers, and of course, the real numbers, aren't mentioned. This is in line with the rest of the book. Halmos even warns the sensitive reader at one point that he might be shocked because the number (e.g. set) 2 is to be used.

The axiom of choice is introduced through the cartesian product, the elements of wich are special functions. This is confusing on a first reading because functions are introduced (before that) as subsets of cartesian products.

This book is fascinating. Halmos proceeds to construct the most relavant concepts of set theory independantly of any other mathematics. For instance never once does he use numbers until he has constructed them out of sets. The level of rigor is not that of axiomatic set theory, so the book is accessible.
Unfortunately, as seems to be Halmos style (definitly evident in his 'Finite Vector Spaces' which I do NOT recommend unless you are far more gifted than I), he is quite compact. He compresses a wealth of information into a very short space, and most of the 25 topics are covered in under 4 full pages. The exercises are sparse and difficult.
This book could definitly have benefited from much more explanation and exercises. For the reader who possess the talent, though, this book is strongly recommended. Even for those (like me) who failed to grasp every detail, it is still a very worthwhile read. I fully intend to return to this when I have a more firm grounding in the thought patterns of abstract mathematics.