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“This book is a very specialized but broadly useful introduction to set theory. It is aimed at ‘the beginning student of advanced mathematics’ … who wants to understand the set-theoretic underpinnings of the mathematics he already knows or will learn soon. It is also useful to the professional mathematician who knew these underpinnings at one time but has now forgotten exactly how they go. … A good reference for how set theory is used in other parts of mathematics … .” (Allen Stenger, The Mathematical Association of America, September, 2011)

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.

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12 of 13 people found the following review helpful

Naive Set Theory, Paul Halmos' classic textbook originally published in 1960 has an accompanying book of Exercises in Set Theory by L. E. Sigler, also published in 1960 (ISBN: 0442780869)

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I remember reading this book back in 1965, as a small part of one of my first year maths courses, and being impressed by the lucid and friendly exposition of the difficult concepts involved in the theory of non-finite sets. Much later, I was privileged to take Paul out to lunch on a visit to our university. He was as interesting in life as in his books. He carried a tiny "spy" camera round his neck to take photos of people he visited!

So this book will take you from the basic operations of union and intersection, through to countable and uncountable sets and the Axiom of Choice, Zorn's Lemma, Transfinite Induction and Well Ordering. It is not a mathematical logic textbook, hence the "Naive" in the title. It will however give you a thorough grounding in the basics, sufficient to understand other subjects where mathematicians prove broad results that apply to both finite and infinite cases, eg, the existence of a basis in any vector space.

A highly recommended little book!

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67 of 69 people found the following review helpful

Mathematical writing at its best10 Jun. 2000

By
"rainbowcrow"
- Published on Amazon.com

Format: Hardcover

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.

17 of 18 people found the following review helpful

The essential essence of set theory in 100 pages21 April 2002

By
Charles Ashbacher
- Published on Amazon.com

Format: Hardcover

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.

14 of 15 people found the following review helpful

Very thorough yet too compact30 July 1999

By
A Customer
- Published on Amazon.com

Format: Hardcover

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.

9 of 9 people found the following review helpful

Not quite perfect27 Mar. 2007

By
Johan Nystrom
- Published on Amazon.com

Format: Hardcover

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!

/***/

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.

16 of 19 people found the following review helpful

2 stars for the automaths, 5 stars for math lovers22 Sept. 2008

By
Joseph Barnett
- Published on Amazon.com

Format: Hardcover

Halmos is a horrible choice for trying to learn set theory on your own. The first chapters of Stoll (he goes way beyond Halmos in scope) are much better for self-teaching. Not that Stoll is 'easier' - hardly - it's just that Halmos isn't trying to teach. His book lacks explanations and examples; the prose is tight and compact, tough to digest. As another reviewer noted, Suppes is a good choice as well.

I read this book in conjunction with Stoll and Suppes - I found myself using Stoll first to understand the subject matter, Suppes to 'shore up' the axiomatic framework, and Halmos last, frankly just to see if I finally understood it. Halmos was my 'test' for understanding. So, no, I don't recommend Halmos for learning the subject.

On the other hand, the book is a classic (I've heard), and a pleasure to read (if you already understand set theory). I would read a page from Halmos, find it painful, learn the material elsewhere, come back, and really enjoy it. I'm glad I own it, but I'm now annoyed with myself because I wrote in it a little before I gave up using it as a textbook.

Also, I'm still trying to understand why he titled the book 'naive.' Obviously it's not a rigorous treatment, yet he covers basic axioms and generalizes the traditional, algebraic, binary set operators to infinite sets... calling it 'naive' doesn't seem correct.