Customer Reviews

Naive Set Theory (Undergraduate Texts in Mathematics)

5 star:		(9)
4 star:		(5)
3 star:		(1)
2 star:		(3)
1 star:		(0)

 

 

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 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.

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.

I beg to differ with other reviewers of this book. Although the book may have been regarded as a classic in its time, it cannot be recommended as a source for either self-study or as a reference book on set theory by present-day standards for mathematical writing. Its style varies from too concise to too verbose in an erratic manner. It attempts to combine rigor and lack of rigor, and does so inconsistently. There are no references to other works. Finally, I was particularly turned off by the last sentence of the Preface, the pompous and patronizing "read it, absorb it, and forget it." Those who are interested in elementary set theory are advised to consult the books by Robert Stoll and Patrick Suppes. Besides being much better written and more comprehensive, they are also cheaper.

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.

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.

I find set theory to be the most intimidating subject in math. It seems so removed, but underpins every assumption I make in mathematics. Many other set theory books are dense and not very clear, but Halmos clearly expounds set theory.

Set theory, as is most mathematics, is hard, so be prepared to think. This book has only 102 pages in it and has just about everything I ever needed to know about set theory for me to feel confident using this theory to understand and prove things in other branches of mathematics.

Halmos's Naive Set Theory is the type of book I look for most, when I'm interested in a topic outside my specialization, but would like to know it better to apply it to my research. It's a clear, concise introduction to set theory, getting to the meat of it, without all the little asides and interesting things that distracts from learning the core of the subject.

This book should be on the bookshelf of every serious (and amateur) mathematician.

this book explains succinctly the basic language of sets, and a few of the more advanced ideas like ordinal numbers. I learned the first half of this stuff in high school, and the other half in college. This book is as good a place to learn it as any, and better than most. Halmos is exactly right when he says "read it, absorb it, forget it", because he is teaching an elementary language. This is set theory for mathematicians, not logicians, so it is explained in words, not tedious logical symbols, intended for people who know the English language.

Everyone doing math should know this much at least by beginning grad school, but there is nothing very deep in it, and once acquired it can be taken for granted, i.e. forgotten consciously. There are some more pedestrian books available, if you prefer them, but this one is excellent for the bright student of high school age or older.

This is still the indispensable introduction to the subject for the student of mathematics, although specialists in logic and set theory will want to dig deeper into the subject. It's style is conversational, yet rigorous and can be either lightly browsed or studied more deeply. Although somewhat dated, it should still be a valuable resource in every mathematician's education.

This book is very clear. The style is informal but the details of the rigor are transparent, which is good for every student of mathematics to see at some time. This is especially important because set theory is something that is often used at the foundation of other mathematical works.

I'm very pleased that a foundations book can be so accessible to undergraduates.