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An Introduction to Godel's Theorems (Cambridge Introductions to Philosophy) [Englisch] [Taschenbuch]

Peter Smith
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26. Juli 2007 Cambridge Introductions to Philosophy
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.

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  • Taschenbuch: 376 Seiten
  • Verlag: Cambridge University Press; Auflage: 1 (26. Juli 2007)
  • Sprache: Englisch
  • ISBN-10: 0521674530
  • ISBN-13: 978-0521674539
  • Größe und/oder Gewicht: 24,4 x 17,3 x 1,8 cm
  • Durchschnittliche Kundenbewertung: 5.0 von 5 Sternen  Alle Rezensionen anzeigen (1 Kundenrezension)
  • Amazon Bestseller-Rang: Nr. 250.615 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
  • Komplettes Inhaltsverzeichnis ansehen

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'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.' Christopher Leary, State University of New York

'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

'… 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 Gödel's theorems, covering many aspects which are not easy to find in other standard presentations.' Mathematical Reviews

Über das Produkt

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.

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5.0 von 5 Sternen A superb book 18. Dezember 2013
Von Leser
Format:Taschenbuch|Verifizierter Kauf
This book is a really great overview on Gödel's Theorems, how to interpret them and how they are connected.

For me the book offers a perfect mix of formality, formulas, proofs on the one side and text and explanation on the other side. The book is not trying to prove the theorems quickly or just elegantly but tries (and succeeds) to explain everything. It is not taking shortcuts in the arguments, it is well written and amusing to read.

This book was not my first encounter with Gödel's theorems but I think I would recommend it also as an introduction for any reader with a mathematical background. So: Simply a great book for anyone.
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Die hilfreichsten Kundenrezensionen auf (beta) 5.0 von 5 Sternen  6 Rezensionen
50 von 50 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen the best thing out there 26. Dezember 2007
Von Jon M. Cogburn - Veröffentlicht auf
Format:Taschenbuch|Verifizierter Kauf
For a couple of decades now, students who had completed their first logic class and dabbled in a little bit of metatheory (perhaps soundness and completeness) were forced to avail themselves of Boolos and Jeffrey's (fourth edition with Burgess) "Computability and Logic." Unfortunately, the third edition presented much of the material in too brief a manner, resulting in a big jump from lower level logic to the material covered. The fourth edition is much longer, but no more easier to teach to talented undergraduates. More recently, Epstein's book on computability was an improvement in this regard, but its logical coverage was much less.

Smith's book should now be the canonical text. First, the discussion and proofs are astoundingly clear to students who haven't done much logic beyond their first class. Pick any topic from B & J and Smith, for example primitive recursiveness, the tie between p.r. axiomatizability and axiomatizability via Craig's theorem, etc. and the discussion and proofs in Smith will be clearer, more accessible, and more clearly tied to the other relevant concepts. Second, the coverage is exactly what is needed to understand both theorems and the most important consequences and extensions. Third, the way he ties the disparate topics together (for example the informal proofs through Chapter 5 and their rigorizaiton through Chapter 18) is just fantastic. This is really important for helping the reader develop a deeper understanding of things. If you just pile theorem upon theorem it's easy for the reader accept them as true without developing any logical insight and appreciation of the landscape.

I don't know if Cambridge would allow this, but in the next edition they should seriously think about adding exercise sections like B & J and Epstein. If they did, I think this book would eclipse the other two for classroom uses.

It's not just for students, either. A colleague and I were arguing about something and we picked up Smith's work rathern than either of Smullyan's to figure out a point relevant to the debate. I find that my grasp of the relevant proofs is much cleaer for reading Smith (my colleague is much, much better at logic than me, but with Smith's help I won the debate).

It is both extraordinary and a cause for celebration when someone can combine in a logic text this level of coverage, rigor, accessibility, and funness of read. I don't think there is a precedent actually. In short, Smith's work is a service to Lady Philosophy. Joe Bob says check it out.
25 von 25 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Good Book, Wrong Title 11. Januar 2009
Von Pedestrian - Veröffentlicht auf
This is a terrific book that the reader can learn a lot from. The author presents Gödel's theorems - in fact, he provides many different proofs of the theorems - along with various strenghtenings and weakenings of the main results. In the many historical and conceptual asides the author does a great job of explaining the significance of Gödel's theorems and of directing the reader's attention to the big picture.

However, I don't think the book is a good *introduction* to Gödel's theorems. A student approaching these theorems for the first time will be overwhelmed by the amount of information here. Even more problematic is the author's adoption of a rather informal way of writing. This does make the book very readable but I think would frustrate the beginning student who needs a precise grasp of new concepts. For example, I don't think a student innocent of primitive recursive functions would be able to grasp how they work from the chapter here. This problem is further compounded by the lack of exercises.

In sum, the book is highly recommended for anyone looking to deepen and broaden their understanding of Gödel's theorems. However, I think that anyone who hasn't already seen a rigorous presentation of those theorems might find the book frustrating.
4 von 4 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Gives some Research Directions 26. April 2011
Von Customer - Veröffentlicht auf
Format:Taschenbuch|Verifizierter Kauf
Mathematical logic in all its dexterity is a somewhat fractured discipline. We have set theory, recursion theory, proof theory, model theory, etc. all falling under the same umbrella. Each of these disciplines move into very sophisticated terrain quickly, sometimes with disparate notation, techniques, goals, etc. Thus it can be a hurdle for one to try to get an overview of each area and their pivotal questions and theorems, despite the claim of their relation to "Mathematical Logic."

The incompleteness theorems of Godel have been an impetus for logical research since their discovery. So it is not out of place for a writer to want to focus on them and give a (relatively) uninitiated audience a survey of relevant results. What is wonderful about Peter Smith's text is how he presents the essentials clearly and touches on how the separate research areas have tried to deal with or explain incompleteness. So in this sense, Smith illustrates at least one way all the areas of mathematical logic are connected, namely how they all have at some point sought to shine light on Godel's results. The fruits of these areas' labors are made clear and accessible in a way that is not duplicated anywhere else; at present, this book is one of a kind.

The only flaw is the lack of exercises, but these can be found in more specialized texts. There is excellent annotation and an accompanying bibliography for those who want to go further.
2 von 2 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Superb ! 29. September 2011
Von André Gargoura - Veröffentlicht auf
All the reviewers are enthusiastic and with good reason.

This is not just one more text on Gödel's theorems, nor just another standard source... : it's "THE" book.

Peter Smith is gifted both as a teacher and as a writer. He managed the "tour de force" of making Gödel's ideas accessible, clearly and uncompromisingly.

Whether it be global architecture, introduction and linkage of basic concepts, footnotes, other sources, text layout... Everything is well thought of , producing a sense of inner beauty.

This book should be read by any would-be writer in mathematics...

If you are really interested in the Gödelian world, buy this book and embark with Peter Smith on a splendid journey.
10 von 16 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen An introduction to Gödel's theorems by Peter Smith 22. Dezember 2007
Von Ricardo C. Martini - Veröffentlicht auf
Format:Taschenbuch|Verifizierter Kauf
Very rigurous but simultaneously very understandable (...for this kind of theorems that have many technical dificulties is much to say...) I enjoy (very much) to read this book (I took it to the beach during my vacations!.)
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