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Sets for Mathematics Hardcover – January 20, 2003

ISBN-13: 978-0521804448 ISBN-10: 0521804442 Edition: 1st

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Editorial Reviews


"...the categorical approach to mathematics has never been presented with greater conviction than it has in this book. The authors show that the use of categories in analyzing the set concept is not only natural, but inevitable." Mathematical Reviews

"To learn set theory this way means not having to relearn it later.... Recommended." Choice

Book Description

Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. For the first time in a text, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically andphysically common phenomena and advancing to a precise specification of the nature of Categories of Sets. An Appendix provides an explicit introduction to necessary concepts from logic, and an extensive Glossary provides a window to the mathematical landscape.
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Product Details

  • Hardcover: 276 pages
  • Publisher: Cambridge University Press; 1 edition (January 20, 2003)
  • Language: English
  • ISBN-10: 0521804442
  • ISBN-13: 978-0521804448
  • Product Dimensions: 6.8 x 0.8 x 9.7 inches
  • Shipping Weight: 1.5 pounds (View shipping rates and policies)
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Best Sellers Rank: #4,199,523 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

28 of 29 people found the following review helpful By P. Waszkiewicz on April 6, 2010
Format: Paperback
There seem to be two types of undergraduate exercises in set theory: the boring ones (e.g. where we are asked to compute the intersection of an indexed family of sets, or draw a diagram of a relational composition) and the exciting ones (where we ask why a set cannot be of the same size as its powerset, or why having two injections f:A->B and g:B->A implies that A and B are isomorphic). The difference is of course that the first kind involves mechanical computation with points, and all data given, and the second kind needs a creative argument in a situation where it sometimes seems that there is not enough data to solve the problem.

The book by Lawvere and Rosebrugh made me realise that the exercises I find exciting can be phrased in terms of properties of maps acting on sets, and - yes, indeed - the boring ones can't.

But then the book goes further - it shows that in fact all axioms of sets can be written down in the language of maps. Doesn't it strike you that as a consequence axiomatic set theory is not boring? :)

Some of the discoveries I made during reading are just invaluable. For example, I learned that the *reason* why a set cannot be of the same size as its powerset is that the two-element set have a self map with no fixed point, which is, admittely, the essence of Cantor's diagonal argument.

The Authors say in the Foreword that the book is for students who are beginning the study of algebra, geometry, analysis, combinatorics, ... Indeed, being a virtuoso of a particular implementation of set theory such as ZFC does not help much with these subjects. Instead one needs a good knowledge of how sets behave when measured, divided, added, towered, counted - name your favourite operation - and this is precisely the story told in the book.
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11 of 12 people found the following review helpful By Michael on July 28, 2010
Format: Hardcover
This book makes the most powerful branch of mathematics accessible to early Mathematics undergraduates. The difficulties people have in comprehending the material are largely due to the abstract nature of the Mathematics itself. I would suggest that a reader takes plenty of time absorbing the material and attempting the exercises. I read Conceptual Mathematics by Dr. Schanuel and Dr. Lawvere before reading this one and I think this helped provide familiarity to the topics discussed. This book contains an ingenious formulation of the familiar Set Theory axioms in the context of the Category of Sets. The Category of Sets here treats all sets with bijections between them as equivalent. A great contribution of one of the authors is the discovery that Cantor adopted this perspective himself. It was Zermelo, who opposed this view, that axiomatized the theory based on the membership relation. This book is a one of a kind read and I would recommend it to ANY sufficiently curious student of Mathematics. I don't claim mastery of all concepts presented, but I do claim many profound insights and a more modern understanding of mathematics. As an undergraduate myself, I can recommend this to other motivated undergraduates with confidence that it is indeed accessible material. It would not surprise me if the content of this book eventually replaced ZFC in university classrooms.
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13 of 15 people found the following review helpful By Christian on February 13, 2011
Format: Hardcover
This should by no means be thought of as a book on traditional set theory. (For that I would recommend "Classic Set Theory" by Goldrei.) This is a book on category theory showing the majority of its interest in the category of sets.

The first few chapters of the book begin to detail a proposed axiomatization of the category of sets, which is finally concluded with the introduction of a natural number object in chapter 9 after being sidetracked for a few chapters by some interesting properties of exponentiation and power sets. This is definitely one of the most interesting mathematics texts I have come across, and I feel like I got a lot out of it.

Despite how deceptively simple the first few exercises were, the difficulty level rocketed up fast and I found the book to be extremely challenging overall. The material itself was hard enough, but the sophistication of the authors' writing only compounded the difficulty. Oftentimes the prose parts of the book felt like something you would see as a "reading comprehension" passage on the GRE, nothing indecipherable, but it certainly took time to process even small bits of content. I quit the book a mere ten pages from the end because I had become completely overwhelmed and was understanding the final material only at a very superficial level.

Someone with a better mind than I have might get a lot out of the things I struggled most with. However, the book does have some errors scattered throughout that cause mild confusion. In addition, there were several points in the book where terminology was invoked that I couldn't recall having read before and couldn't find in the index of terms.
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