Customer Reviews: Read 8 more reviews...
 the standard place to start, and a good one November 13, 2008
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.
 2 stars for the automaths, 5 stars for math lovers September 22, 2008
1 out of 1 found this review helpful
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.
 Naive? just the publisher's "come on" first class work July 1, 2007
1 out of 3 found this review helpful
Quite a good introduction to set theory, ideal for the person too busy
to read all the latest texts produced by academia. Also for the person
scared by too much mathematical symbolism, makes them realise thatwhile
symbolism is an aid in math it is not of course absolutely necessary.
In physics we see the current downgrading of quantum physics, so we
we should see a reduction in "math trash" as a consequence.
So irrespective of the background, all who read this book should be
rewarded with a greater grasp of math and how it is developed.
Not quite perfect March 27, 2007
4 out of 4 found this review helpful
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.
The Classic Introduction to Set Theory January 4, 2007
2 out of 2 found this review helpful
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.
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