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Set Theory and the Continuum Hypothesis (Dover Books on Mathematics)

4.1 out of 5 stars 12 customer reviews
ISBN-13: 978-0486469218
ISBN-10: 0486469212
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Editorial Reviews

About the Author

A renowned mathematician, professor, and theorist, the late Paul J. Cohen won two of the most prestigious awards in mathematics: the American Mathematical Society's Bôcher Prize in 1964, for analysis; and the Fields Medal, the "Nobel Prize" of mathematics, in 1966, for logic.
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 190 pages
  • Publisher: Dover Publications (December 9, 2008)
  • Language: English
  • ISBN-10: 0486469212
  • ISBN-13: 978-0486469218
  • Product Dimensions: 6.1 x 0.4 x 9.1 inches
  • Shipping Weight: 8.8 ounces (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #343,450 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

Format: Paperback Verified Purchase
Paul Cohen's "Set Theory and the Continuum Hypothesis" is not only the best technical treatment of his solution to the most notorious unsolved problem in mathematics, it is the best introduction to mathematical logic (though Manin's "A Course in Mathematical Logic" is also remarkably excellent and is the first book to read after this one).
Although it is only 154 pages, it is remarkably wide-ranging, and has held up very well in the 37 years since it was first published. Cohen is a very good mathematical writer and his arrangement of the material is irreproachable. All the arguments are well-motivated, the number of details left to the reader is not too large, and everything is set in a clear philosophical context. The book is completely self-contained and is rich with hints and ideas that will lead the reader to further work in mathematical logic.
It is one of my two favorite math books (the other being Conway's "On Numbers and Games"). My copy is falling apart from extreme overuse.
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Format: Paperback
This is still the definitive work on set theory and the continuum hypothesis. Although extremely terse, it is wonderfully clear and unburdened by the technical and pedantic details that doom many books in the subject. If you cannot track this down right now be patient, the American Mathematical Society is going to be reprinting it.

Professor Cohen passed away in March of 2007, but thankfully this book remains as a testament to his genius. Originally trained as an analyst, he began working on the continuum hypothesis knowing almost nothing about logic or set theory. Within two years he mastered the subject and solved the greatest outstanding problem in the field (and arguably in all of mathematics). Read this book if you want to understand one of the deepest ideas in all of human thought.
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As a work of science, "Set Theory and the Continuum Hypothesis" stands on a par with Darwin's "On the Origin of Species". First, like Darwin's book, Cohen's work is a profound contribution to its field; second it is also accessible to any educated and interested reader, although with some effort.

This edition is a reproduction of the first edition. You might be shocked by the type-this is a plain, typewritten document with no illustrations (I find it charming)-but Paul Cohen's crystal clear prose makes the book eminently readable.

=WHAT YOU NEED=

This is a graduate level book but you don't need to be a graduate student in mathematics to understand it. You do need a laymen's interest in mathematics; for instance you should enjoy reading Euclid, Ian Stewart, Douglas Hoftstadter, Martin Gardner. If you've enjoyed Douglas Hofstadter's "Gödel, Escher, and Bach" then there is no reason you can't understand this book.

=WHAT IT DELIVERS=

First, Cohen gives a barebones but complete introduction to formal logic and logical notation.

Then he describes formal set theory, known as Zemerlo Frankel set theory, the foundation of all mathematics as it stands today.

Having spent half the book on the necessary background, Cohen arrives to his main topic, the Continuum Hypothesis and whether it is true or false.

=WHAT COHEN SAYS=

ST&CH proves that a long standing problem in mathematics (the Continuum Hypothesis) has no solution. What does this mean?

Most mathematicians believe in a scaled down version of Hilbert's Programme. Hilbert hoped that all of mathematics followed from a small collection of definitions and axioms, much like all of geometry was once believed to follow from Euclid's five axioms.
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Format: Paperback
Paul Cohen invented forcing as a method for proving the independence of the Continuum Hypothesis from the axioms of set theory, but his approach to the independence proof--which is presented in the last chapter of this book--has been superseded by other approaches (e.g. boolean-valued models) that are easier to follow and more mathematically enlightening. Still, I would highly recommended this little book for two reasons. First, its rapid traversal of first-order logic--completeness, compactness, etc.--is wonderfully lucid and concise: a real mathematician's treatment, blessedly free of the logician's "hair." The same is true of his treatment of axiomatic set theory. His proof, for example, of the Schroeder-Bernstein theorem (if there is an injection from A into B and from B into A, then there is a bijection between A and B) is the briefest and most elegant (and most intuitive) I've ever seen--about three lines, as compared with the usual three-quarters of a page. The second thing I enjoyed about this book is Cohen's many fascinating obiter dicta--for example, when he conjectures near the end that the continuum, being "an incredibly rich set given to us by a bold new axiom [the power-set axiom]," may well have a cardinality that outstrips any aleph obtainable from the replacement axiom. Cohen is as fresh and intuitive as he is intrepid.
The volume has very interesting prefatory memoir by Cohen in which he recounts how he was attracted to the Continuum Problem and his meetings with Kurt Gödel after he (Cohen) obtained his independence proof. There is also a nice introduction by the great logician Martin Davis.
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