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Naive Set Theory (Undergraduate Texts in Mathematics) Hardcover – October 24, 1974

ISBN-13: 978-0387900926 ISBN-10: 0387900926 Edition: 1974th

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Product Details

  • Series: Undergraduate Texts in Mathematics
  • Hardcover: 104 pages
  • Publisher: Springer; 1974 edition (October 24, 1974)
  • Language: English
  • ISBN-10: 0387900926
  • ISBN-13: 978-0387900926
  • Product Dimensions: 9 x 6 x 0.3 inches
  • Shipping Weight: 11.2 ounces (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (25 customer reviews)
  • Amazon Best Sellers Rank: #671,842 in Books (See Top 100 in Books)

Editorial Reviews


From the reviews:

“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)

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Customer Reviews

This is, quite simply, a beautiful book.
I'm very pleased that a foundations book can be so accessible to undergraduates.
Paul Halmos's book is the best introductory text to set theory.

Most Helpful Customer Reviews

67 of 68 people found the following review helpful By "rainbowcrow" on June 9, 2000
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.
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17 of 18 people found the following review helpful By Charles Ashbacher HALL OF FAMETOP 500 REVIEWERVINE VOICE on April 21, 2002
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.
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14 of 15 people found the following review helpful By A Customer on July 30, 1999
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.
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9 of 9 people found the following review helpful By Johan Nystrom on March 27, 2007
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.
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