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Cambridge Introductions to Philosophy Ser.: Introduction to Gödel's Theorems by Peter Smith (2007, Perfect)
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About this product
Product Identifiers
PublisherCambridge University Press
ISBN-100521674530
ISBN-139780521674539
eBay Product ID (ePID)59035811
Product Key Features
Number of Pages376 Pages
Publication NameIntroduction to Gödel's Theorems
LanguageEnglish
Publication Year2007
SubjectLogic
TypeTextbook
AuthorPeter Smith
Subject AreaMathematics
SeriesCambridge Introductions to Philosophy Ser.
FormatPerfect
Dimensions
Item Height0.9 in
Item Weight26.5 Oz
Item Length9.8 in
Item Width6.9 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN2007-299862
TitleLeadingAn
Dewey Edition22
Reviews"... 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 Gdel's theorems, covering many aspects which are not easy to find in other presentations." Reinhard Kahle, Mathematical Reviews, 'Peter Smith has succeeded in writing an excellent introduction to G'del'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 G'del's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, Department of Philosophy, University of Calgary, 'Smith has written a wonderful book giving a clear and compelling presentation of Gdel'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 Gdel'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, 'Smith has written a wonderful book giving a clear and compelling presentation of Gödel'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 Gödel'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.'Professor Christopher Leary, Department of Mathematics, SUNY Geneseo, '… 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 Gdel's theorems, covering many aspects which are not easy to find in other standard presentations.' Mathematical Reviews, 'Peter Smith has succeeded in writing an excellent introduction to G del'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 G del's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, University of Calgary, 'Smith has written a wonderful book giving a clear and compelling presentation of G�'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 G�'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.' Professor Christopher Leary, Department of Mathematics, SUNY Geneseo, 'Peter Smith has succeeded in writing an excellent introduction to Gödel'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 Gödel's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, University of Calgary, 'Smith has written a wonderful book giving a clear and compelling presentation of G'del?'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 G'del?'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.' Professor Christopher Leary, Department of Mathematics, SUNY Geneseo, "How did Gödel 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 Gdel'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, 'Peter Smith has succeeded in writing an excellent introduction to Gödel'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 Gödel's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, Department of Philosophy, University of Calgary, "How did Gdel 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 Gdel'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, 'Smith has written a wonderful book giving a clear and compelling presentation of Gödel'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 Gödel'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.' Professor Christopher Leary, Department of Mathematics, SUNY Geneseo, 'Peter Smith has succeeded in writing an excellent introduction to G�'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 G�'s results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, Department of Philosophy, University of Calgary, 'Smith has written a wonderful book giving a clear and compelling presentation of Gödel’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 Gödel’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.'Professor Christopher Leary, Department of Mathematics, SUNY Geneseo, "... 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 Gödel's theorems, covering many aspects which are not easy to find in other presentations." Reinhard Kahle, Mathematical Reviews, 'Smith has written a wonderful book giving a clear and compelling presentation of G del'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 G del'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 Gdel'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 Gdel's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, University of Calgary
Dewey Decimal511.3
Table Of ContentPreface; 1. What Gödel's Theorems say; 2. Decidability and enumerability; 3. Axiomatized formal theories; 4. Capturing numerical properties; 5. The truths of arithmetic; 6. Sufficiently strong arithmetics; 7. Interlude: taking stock; 8. Two formalized arithmetics; 9. What Q can prove; 10. First-order Peano Arithmetic; 11. Primitive recursive functions; 12. Capturing funtions; 13. Q is p.r. adequate; 14. Interlude: a very little about Principia; 15. The arithmetization of syntax; 16. PA is incomplete; 17. Gödel's First Theorem; 18. Interlude: about the First Theorem; 19. Strengthening the First Theorem; 20. The Diagonalization Lemma; 21. Using the Diagonalization Lemma; 22. Second-order arithmetics; 23. Interlude: incompleteness and Isaacson's conjecture; 24. Gödel's Second Theorem for PA; 25. The derivability conditions; 26. Deriving the derivability conditions; 27. Reflections; 28. Interlude: about the Second Theorem; 29. Recursive functions; 30. Undecidability and incompleteness; 31. Turing machines; 32. Turing machines and recursiveness; 33. Halting problems; 34. The Church-Turing Thesis; 35. Proving the Thesis?; 36. Looking back.
SynopsisIn 1931, the young Kurt G'del 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. G'del 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., In 1931, the young Kurt Gödel 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. Gödel 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., What are Gödel's Theorems, how were they established and why do they matter? Written with great clarity, this book is accessible to philosophy students with a limited formal background. It is equally valuable to mathematics students taking a first course in mathematical logic., In 1931, the young Kurt Godel 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. Godel 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.
LC Classification NumberQA9.65
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