Naive set theory (The University series in undergraduate mathematics) (Hardcover)

by Paul R Halmos (Author) "A pack of wolves, a bunch of grapes, or a flock of pigeons are all examples of sets of things..." (more)
Key Phrases: transfinite recursion theorem, diminished set, ordinal arithmetic (more...)

Editorial Reviews

Product Description
From the Reviews:

"...He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know. ...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics." Philosophy and Phenomenological Research --This text refers to the Hardcover edition.

Product Details

• Hardcover: 104 pages
• Publisher: Van Nostrand (1968)
• ASIN: B0007FR8J4
• Average Customer Review: 4.2 out of 5 stars See all reviews (16 customer reviews)
• Amazon.com Sales Rank: #7,061,025 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

38 of 40 people found the following review helpful:

5.0 out of 5 stars Mathematical writing at its best, June 9, 2000

By	"rainbowcrow" (Seattle, WA United States) - See all my reviews

This review is from: Naive Set Theory (Undergraduate Texts in Mathematics) (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.

12 of 12 people found the following review helpful:

4.0 out of 5 stars The essential essence of set theory in 100 pages, April 21, 2002

By	Charles Ashbacher "([e-mail])" (Marion, Iowa United States([e-mail])) - See all my reviews (TOP 50 REVIEWER)

This review is from: Naive Set Theory (Undergraduate Texts in Mathematics) (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.

16 of 18 people found the following review helpful:

2.0 out of 5 stars A contrarian opinion, April 3, 2006

By	Reviewer from Palo Alto (Palo Alto, California United States) - See all my reviews

This review is from: Naive Set Theory (Undergraduate Texts in Mathematics) (Hardcover)

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.