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An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy)

An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy)

by PeterSmith (Author)

  • Used

    Paperback 2007

    $31.17
Synopsis

In 1931, the young Kurt Goedel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Goedel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.

$31.17

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More Information

Format: Paperback
 Pages: 376
 Edition: 1
 Publisher: Cambridge University Press
Published: 26 Jul 2007
ISBN 10: 0521674530
ISBN 13: 9780521674539
Book Overview: Peter Smith examines Goedel's Theorems, how they were established and why they matter.

Media Reviews
'Smith has written a wonderful book giving a clear and compelling presentation of Goedel's Theorems and their implications. His style is both precise and engaging at the same time. The clarity of the writing is impressive, and there is a pleasing coverage of historical and philosophical topics. An Introduction to Goedel's Theorems will work very well either as a textbook or as an introduction for any reader who wants a thorough understanding of some of the central ideas at the intersection of philosophy, mathematics and computer science.' Christopher Leary, State University of New York
'Peter Smith has succeeded in writing an excellent introduction to Goedel's incompleteness theorems and related topics which is accessible without being superficial. Philosophers in particular will appreciate the discussions of the Church-Turing Thesis, mechanism, and the relevance of Goedel's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, University of Calgary
'... it is, without doubt, a mandatory reference for every philosopher interested in philosophy of mathematics. The text is, in general, written in a prose style but without avoiding formalisms. It is very accurate in the mathematical arguments and it offers to mathematicians and logicians a detailed approach to Goedel's theorems, covering many aspects which are not easy to find in other standard presentations.' Mathematical Reviews
How did Goedel establish the two Theorems of Incompleteness, and why do they matter? Smith (U. of Cambridge) advises readers to take their time in answering these and related questions he poses as he presents a variety of proofs for the First Theorem and shows how to prove the Second. He also examines a group of related results with the same care and attention to detail. In 36 well-paced chapters Smith builds his case from a basic introduction to G<:o>del's theorems on to such issues as the truths of arithmetic, formalized arithmetics, primitive recursive functions, identifying the diagonalization Lemma in the First Theorem and using it, dirivability conditions in the Second Theorem. Turing machines (and recursiveness) and the Church-Turing thesis. Accessible without being dismissive, this is accessible to philosophy students and equally suitable for mathematics students taking a first course in logic. Book News
... Without doubt, a mandatory reference for every philosopher interested in philosophy of mathematics. The text is, in general, written in a prose style but without avoiding formalisms. It is very accurate in the mathematical arguments and it offers to mathematicians and logicians a detailed approach to Goedel's theorems, covering many aspects which are not easy to find in other presentations. Reinhard Kahle, Mathematical Reviews
Author Bio
Peter Smith is Lecturer in Philosophy at the University of Cambridge. His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003), and he is a former editor of the journal Analysis.