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Cambridge Introductions to Philosophy
An Introduction to Gödel’s Theorems
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.ÌýHow are these Theorems established, and why do they matter?Ìý 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.
402 pages, Paperback
First published January 1, 2007
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About the author
Peter Smith
358Ìýbooks21ÌýfollowersLibrarian Note: There is more than one author in the GoodReads database with this name.
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Displaying 1 - 5 of 5 reviews
June 20, 2013
I stumbled on this book and decided to give it a try. I'm glad I did. I've been working my way through it for a while now. It sits between texts for general audiences that are short on details, and mathematical texts that provide little to no explanatory text. In this book you get the proofs and the discussion both.
You get pretty general proofs of incompleteness early on and then spend many pages filling in the details. After some discussion of possible misinterpretations and stronger forms of the first theorem he covers related results such as those from Lob and Turing. I haven't read all of the later chapters that go beyond Godel's theorems but what I have read seem to be of the same high quality as the rest of the book.
It could have used a logic primer but I think this particular book series has length restrictions. Plus, the author has written an introduction to logic book as well so one can try to get their hands on that. If you are a layman and want to know more about the incompleteness theorems you can't really go wrong with this book. It is easily the best I have seen in this category.
You get pretty general proofs of incompleteness early on and then spend many pages filling in the details. After some discussion of possible misinterpretations and stronger forms of the first theorem he covers related results such as those from Lob and Turing. I haven't read all of the later chapters that go beyond Godel's theorems but what I have read seem to be of the same high quality as the rest of the book.
It could have used a logic primer but I think this particular book series has length restrictions. Plus, the author has written an introduction to logic book as well so one can try to get their hands on that. If you are a layman and want to know more about the incompleteness theorems you can't really go wrong with this book. It is easily the best I have seen in this category.
January 23, 2019
4.5/5.0
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September 24, 2020A mental trip. I understood 30% max...
February 15, 2025
Peter Smith is a great teacher. The material here is distilled, organized and clear.
January 3, 2013
Very Bstract, dense, but if you stick to it, you will reap the benefits !!
Displaying 1 - 5 of 5 reviews
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