Godel's Incompleteness Theorems (Oxford Logic Guides) [Hardcover]

Raymond M. Smullyan (Author)

Book Description

ISBN-10: 0195046722 | ISBN-13: 978-0195046724 | Publication Date: August 20, 1992 | Edition: First

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

Editorial Reviews

Review

"A delightful introduction to the Godel incompleteness theorems and related results. . . . reader is treated to a series of careful formulations and proofs of the central results, always with a high degree of generality and insight. The author has superbly combined his unique presentation of the 'big picture' with an appreciation of detail and rigor. Even readers who are already familiar with the incompleteness results will enjoy and benefit from this book." --Mathematical Reviews

"Elegant . . . the strategy . . . is highly instructive, as it casts the purpose behind each step of the proof in high relief." --Choice

"Combines scholarly contributions with the flavor of his popular works. Smullyan is not only an outstanding authority on this subject, but also a skilled pedagogue, with a special talent for formulating simple riddles, which illuminate this very difficult and profound subject. . . . an important contribution toward the wider understanding of the work of Godel and his followers. . . . Smullyan plays a significant role in the further development of mathematical logic and the elucidation of its relation to metamathematics. He continues to be one of the foremost popularizers of the subject." --American Scientist

"Smullyan lives up to his aims. The book provides a highly accessible, user-friendly introduction to incompleteness. . . . the treatment is rigorous and contains material that even a professional logician can find informative and interesting. . . . Smullyan never confuses rigor with dullness or obscurity. His writing is clear and lively . . . . I am eagerly awaiting the sequel's appearance." --Leon Harkleroad, Modern Logic

About the Author

Raymond M. Smullyan is at Indiana University, Bloomington.

Product Details

• Hardcover: 160 pages
• Publisher: Oxford University Press, USA; First edition (August 20, 1992)
• Language: English
• ISBN-10: 0195046722
• ISBN-13: 978-0195046724
• Product Dimensions: 9.3 x 6.3 x 0.6 inches
• Shipping Weight: 13.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #895,649 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

35 of 36 people found the following review helpful:

5.0 out of 5 stars Mainline Incompleteness with this Book!, March 22, 2000

By A Customer

This review is from: Godel's Incompleteness Theorems (Oxford Logic Guides) (Hardcover)

I highly recommend this title because it supplys all the necessary proofs for a nuts and bolts understanding of incompleteness, including incompleteness proofs for Peano arithmetic and the unprovability of consistency.

This title is a difficult read but the only prerequisite is a familiarity of first-order logic equivalent to a one semester college course.

A lot of the proofs are based on new material and are easier to understand than the original work by KG.

An added benefit is the exercises. They are not impossible and aid in one's understanding.

This book is well worth the work in demands.

20 of 21 people found the following review helpful:

5.0 out of 5 stars A very good book but requires correction of typos, July 5, 2006

By 

Ng, Yui Kin (Hong Kong) - See all my reviews
(REAL NAME)   

This review is from: Godel's Incompleteness Theorems (Oxford Logic Guides) (Hardcover)

Raymond Smullyan is a logician that I admire much. This book is very good but contains many typos and mistakes. For example, in p.31, the definition of xPy does not work as it is not able to account for 0P305. The definition should be corrected as:
xPy iff There is z not greater than y (zBy and xEz). Similar mistakes can be found elsewhere. And this book thus requires another edition for the coorection of typos and mistakes.

33 of 39 people found the following review helpful:

5.0 out of 5 stars Finally -- Straight Talk About Incompleteness!, August 2, 2001

By 

James R. Mccall (Libertyville, IL USA) - See all my reviews
(REAL NAME)   

This review is from: Godel's Incompleteness Theorems (Oxford Logic Guides) (Hardcover)

Well. This is the book. Read this instead of, or before you read Goedels paper. Within 20 pages you will know the trick that Goedel used. Its a beauty, but it is far easier to see it under Smullyans tutelage than by coming to the classic paper cold, since Goedel uses a more difficult scheme to achieve his ends. Much work has been done since 1931, and we get the benefit of the stripping-down to essentials that such as Tarski (and Smullyan himself) have contributed.

The book has much of interest to those who wish to pursue the subject of the incompleteness and/or consistency of mathematics, or to come at Goedel from a number of angles. For me, though, the first 3 chapters were enough. I just wanted to find out how K.G. did what he did. Now I know, and I know where to go if I need even more.

The exercises are helpful to keep you on track and test your understanding. They also contribute materially to the exposition. A stumbling-block for many readers will be the extremely abstract nature of the discussion, and the new notations and definitions that constantly come at one. Viewing numbers as strings and strings as numbers (and knowing when to switch from one view to another) will be confusing at first. This is the hard part: what Goedel did, in essence, is demonstrate that one can view proofs in two ways  as numbers, and as strings of characters. As in viewing an optical illusion, it is sometimes tough to hold the proper picture in mind.

Smullyans book First-Order Logic is enough preparation for this work. One must here, even more than there, keep straight the difference between the proofs that are part of the subject matter (and so are strings of characters), and the proofs we go through that verify facts about these strings. Before we started reading this book, of course, we had some informal sense that we were going to prove something about proofs. What we are REALLY doing, though, is proving something about proofs. You get the picture. Goedel must have been a lot of fun at parties.