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Gödel's Proof
In 1931 Kurt Godel published his fundamental paper, "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Godel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement i
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Hardcover, 129 pages
Published
October 1st 2001
by New York University Press
(first published 1958)
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This Statement is False.
How can anyone wrap oneself around that ancient paradox? A true statement whose unprovability resulted precisely from its truth!
With this self-undermining formula as his hammer, Godel knocked down the fortress of Principia Mathematica (PM), so painstakingly constructed by Russell and Whitehead.
He also showed that his method applied to any system whatsoever that tried to accomplish the goals of Principia Mathematica. In effect, then, Godel destroyed the hopes of those who ...more
How can anyone wrap oneself around that ancient paradox? A true statement whose unprovability resulted precisely from its truth!
With this self-undermining formula as his hammer, Godel knocked down the fortress of Principia Mathematica (PM), so painstakingly constructed by Russell and Whitehead.
He also showed that his method applied to any system whatsoever that tried to accomplish the goals of Principia Mathematica. In effect, then, Godel destroyed the hopes of those who ...more
My thanks to AC for convincing me to take the plunge and purchase this little gem: who'd have thought that one-hundred-and-thirteen pages of mathematical logic could have been so entertainingly informative? It was little touches like the chess analogy for describing the relationship between mathematics and metamathematics, the placing of the Richard paradox in terms that were more pellucid than the valiant effort attempted by Rebecca Goldstein, and the tricky, but effective, explanation of how G
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Ce livre comporte trois ouvrages distincts. Premièrement un texte de vulgarisation visant à présenter le théorème d'incomplétude de Kurt Gödel, et à expliciter autant que possible pour le tout venant comme moi les grandes étapes du raisonnements, ainsi que les techniques employées par Gödel. C'est pas mal, les notions sont introduites par degré, les grandes étapes sont exposées, ainsi que les techniques employées. Il me semble que l'on vise un public de philosophes plutôt que de matheux.
En secon ...more
En secon ...more
I don't read much math these days, so when I do read it, it's a little like climbing a steep wall following a winter of sitting in front of a computer. You discover a lot of soreness in muscles you scarcely remember you ever had in the first place.
The long and the short of it is that once upon a time, I sorta understood Godel's incompleteness theorem, and after this modest reading, I sorta understand it again. Success!
More importantly for me, it was fun to try to connect neurons in my poor fuzzy ...more
The long and the short of it is that once upon a time, I sorta understood Godel's incompleteness theorem, and after this modest reading, I sorta understand it again. Success!
More importantly for me, it was fun to try to connect neurons in my poor fuzzy ...more
This book is one of those rare creations in which its clarity and succinctness of presentation highlights the most important concepts. Even if one is not interested in the theory itself the first half of the book is a must read by anyone dealing with mathematics or interested in the nature of truth. It ultimately describes the 1931 paper Kurt Godel published in German entitled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”. The “Principia Mathematica” was the
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This was an extremely difficult book for me. I understood so little of it on my own but luckily was reading it in school where I had people around me who were more mathematically fluent than myself to explain what the hell was going on. It was fascinating and frustrating and the basic ideas I gleaned from it were worth the headaches
Excellent explication of Godel's proof. I appreciate both the simplicity and accuracy of the account this book gives, and the fact that it does not take Godel and make ridiculous assertions about what is suggested by his conclusions, using Godel to endorse a vague mysticism or intuitionism. Wonderful book.
The Book is the best to explain Godel's Proof of the Incompleteness Theorem.
Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true arithmetical statements that cannot be derived from the set.
Godel showed that it is impossible to give a meta-mathematical proof of the consistency of a system comprehensive enough to contain the whole of arithmetic—unless the ...more
Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true arithmetical statements that cannot be derived from the set.
Godel showed that it is impossible to give a meta-mathematical proof of the consistency of a system comprehensive enough to contain the whole of arithmetic—unless the ...more
I recommend this book for readers who want a clear and concise introduction to Godel's proof. The book will be especially useful for readers whose interests lie primarily in mathematics or logic, but who do not have very much prior knowledge of this important proof. Readers with broader interests, who would like to explore the larger implications of the proof for science or philosophy, may be disappointed that the book ends where it does. Godel's Incompleteness Theorem is cited by many scholars
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Once there was a boy, a newcommer to university, who wished to construct a logical paradigm to explain everything in a crystal clear rational proposeitional structure. Then, when he seemed to be mostly achieved his goal, this book came out of nowhere and learned him the humbleness essential for an incomplete mind. It is a brief, very easy to read, and even pop/ordinary book. But I am deeply in debt of it, because I knew Godel through this, and he changed my life...
If you have watched the movie “Inception” and thought it was not bad, then there is a higher chance that you will like this book. I would also give this book another name: “Gödel’s Trick in two parts, Part 2: proof by inception”. An imaginary “Part 1” refers to what has been omitted, i.e. the proof of why the correspondence between activities in different levels of inception is guaranteed to be valid. However, this point does not at all undermine the fact that this is a great and fun book, and a
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This book is about a revolutionary mathematical paper by Kurt Godel. Godel showed that general mathematical statements could be represented using plain ol' numbers, and through this statement-to-number mapping proved that there will always exist an infinite number of true mathematical statements that are not provable using mathematical reasoning, no matter how many new rules one adds. This holds within any axiomatic system which encompasses the whole of number theory.
The book dumbs down the proo ...more
The book dumbs down the proo ...more
Gödel was, essentially, able to prove that any completely logical system of thought of sufficient power had true propositions that necessarily and fundamentally could not be proven--AND that there was no intellectual trick that extend the power of such as system to exclude that outcome. It is one of the great limitative results of twentieth century thought, in some ways similar to Heisenberg's Uncertainty Principle.
This monograph is highly recommended IF you are very interested in the foundation ...more
This monograph is highly recommended IF you are very interested in the foundation ...more
'Unsatisfying' is the best way to describe my experience with this book. The main goal of the book was to relieve some gloomy misconceptions about what the theorem had implied for the future of artificial intelligence in the first half of 20th century. The issue has lost its edge since, perhaps, in part due to popular accounts like this, and certainly due to the fact that a lot of what was "future" of computers in the 1950s has already played itself out. The result was a very practical, even if
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Até a parte preliminar à explicação da prova de Gödel eu entendi, já quanto a prova em si... nãããããooo.
Mas tenho algumas observações.
1ª. **Consistência de um sistema**.
Dado um sistema formado por um conjunto finito de axiomas, provando-se que esse sistema jamais produzirá teoremas contraditórios, então o sistema é consistente.
2ª. No fim das contas (que contas!) a prova de Gödel mostra que, se um sistema é consistente ele terá algumas proposições verdadeiras que não podem ser demostradas nem nega ...more
This book was written by Ernest Nagel, a professor of mathematics and philosophy, and James Newman, the editor of the 4 volume The World of Mathematics. It is aimed at non-mathematicians, but provides a detailed and very understandable walk-through of Godel's 1931 paper on the incompleteness of any axiomatic system that is powerful enough to represent the basic structure of arithmetic.
This is a fascinating book. Having labored through the original proof of the incompleteness theorem years ago, I ...more
This is a fascinating book. Having labored through the original proof of the incompleteness theorem years ago, I ...more
Nothing super-illuminating here
Lots of background on Gödel's incompleteness proof, which was part of the reason I was a little disappointed by Incompleteness, by Rebecca Goldstein. This was a more mathematical treatment than hers, but probably about the same as in Gödel, Escher, Bach: An Eternal Golden Braid. I did appreciate the fact that they covered the Gödel numbering (or at least a fairly close and simplified version of it) to allow the reader to feel assured that a unique formula of Princi ...more
Lots of background on Gödel's incompleteness proof, which was part of the reason I was a little disappointed by Incompleteness, by Rebecca Goldstein. This was a more mathematical treatment than hers, but probably about the same as in Gödel, Escher, Bach: An Eternal Golden Braid. I did appreciate the fact that they covered the Gödel numbering (or at least a fairly close and simplified version of it) to allow the reader to feel assured that a unique formula of Princi ...more
این کتاب توضیحیست بر اثبات قضیایای ناتمامیت اوّل و دوم کورت گودل، ریاضیدان، منطقدان و فیلسوف اتریشی. قضیایای که به نوعی در ریاضیات و منطق انقلابی به شمار میآیند:
قضیهی ناتمامیت اوّل گودل بیان میکند نمیتوان هیچ نظام اصل موضوعی پایداری در ریاضیات بنا گذاشت که در آن بتوان از تعدادی اصل موضوع معدود، تمام حقیقتهای درست ریاضی را استخراج کرد یا خلاف آنها را اثبات کرد و در هر نظام اصل موضوعی بینهایت گزارهی "تصمیم ناپذیر" یافت میشود.
در ناتمامیت دوّم اثبات میشود پایداری هیچ نظامی را نمیتوان در خود آن اثبات ...more
قضیهی ناتمامیت اوّل گودل بیان میکند نمیتوان هیچ نظام اصل موضوعی پایداری در ریاضیات بنا گذاشت که در آن بتوان از تعدادی اصل موضوع معدود، تمام حقیقتهای درست ریاضی را استخراج کرد یا خلاف آنها را اثبات کرد و در هر نظام اصل موضوعی بینهایت گزارهی "تصمیم ناپذیر" یافت میشود.
در ناتمامیت دوّم اثبات میشود پایداری هیچ نظامی را نمیتوان در خود آن اثبات ...more
This little book represents one of the clearest examples of excellent exposition I think I've ever encountered as a young academic. Godel's proof is notoriously difficult to understand, and yet this text manages to deal with the proof in almost predominantly conceptual language. Thoroughly enjoyed the clarity and concision, although now I want to read a more technical presentation to see if I really get it. The content, Godel's famous incompleteness theorem, is always fascinating. I highly encou
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a wonderful and concise look at the prehistory, main concepts and implications of goedel's 1931 proof, detailing how he used a complex mapping system (almost like talking with integers, still mind-boggling to me in a time when computers are commonplace) to blur the line between mathematics and meta-mathematics and reveal irreparable holes in systems of deduction. i like it on page 103 when nagel and newman, apparently bored with writing the words "proof" and "theorem", refer to goedel's "recipe"
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Gödel's incompleteness theorem states that any consistent system of axioms is incomplete i.e. not all theorems in the system can be traced down to the axioms. Moreover, it proves that any consistent system of axioms is also "essentially" incomplete i.e. an attempt to augment the system with further axioms cannot convert it to a complete system.
The book tries to simplify Gödel's proof of incompleteness to some extent, but the details are still hard to digest. The essence of the proof is however w ...more
The book tries to simplify Gödel's proof of incompleteness to some extent, but the details are still hard to digest. The essence of the proof is however w ...more
This is the book that inspired Douglas Hofstadter to write Gdel, Escher, Bach, as he tells us in the prefaces to both books. This edition is Hofstadter's "revision" of the original 1938 book; I'm not sure what if anything he changed beyond adding the preface and three unimportant footnotes marked "--ed." Interestingly, the same year he revised this book by Nagel and Newman he issued GEB unchanged because he didn't feel he had the "right" to revise a book by his "earlier self."
The book is a clear ...more
The book is a clear ...more
A wonderful little book about the complete upending of mathematics which a 25 year-old Austrian logician achieved. The book is difficult, but patient, as it walks the non-expert (read: me) through Godel's gargantuan proof. Using the concept of paradoxes ("This statement is false"), Godel proves that a consistent formal system can never be complete. This simple thesis which he painstakingly showed, destroyed the hopes of mathematicians (chief among them Bertrand Russell and Alfred North Whitehead
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I caveat this short review stating that I will probably have to read this book a couple of more times before giving a final judgement.
However, two heated comments.
Under a stylistic point of view, as the Nature quote in the back of the book states, this text is simply "a little masterpiece of exegesis". Beautifully written, simple enough (although simple is not the right word for one of the most mind blowing concepts in mathematical philosophy), and balanced between abstract analysis of the outco ...more
However, two heated comments.
Under a stylistic point of view, as the Nature quote in the back of the book states, this text is simply "a little masterpiece of exegesis". Beautifully written, simple enough (although simple is not the right word for one of the most mind blowing concepts in mathematical philosophy), and balanced between abstract analysis of the outco ...more
Fundamental principles are important to understand because they have a broad impact beyond their immediate field. Think of Einstein's proof of general relativity.
Godel's proof has similar importance but less popularity. Its direct application is to number theory but is used by systems based on numbers, such as computers.
The book is a fine explanation of the proof. The only reason I wouldn't recommend the book to many people is that videos do a better job of explaining these difficult concepts to ...more
Godel's proof has similar importance but less popularity. Its direct application is to number theory but is used by systems based on numbers, such as computers.
The book is a fine explanation of the proof. The only reason I wouldn't recommend the book to many people is that videos do a better job of explaining these difficult concepts to ...more
A fantastic edited edition of the original by Nagel and Neumann.
The examples are simple to understand, yet deep in their meanings and implications. The footnotes were helpful, although extremely cumbersome in some pages, and overall it was a pleasurable read.
You don't require any background in logic, although more familiar readers should find it a helpful bridge to more advanced logic texts.
Overall, a book which requires more then one read to scratch the surface, and a lifetime to understand it' ...more
The examples are simple to understand, yet deep in their meanings and implications. The footnotes were helpful, although extremely cumbersome in some pages, and overall it was a pleasurable read.
You don't require any background in logic, although more familiar readers should find it a helpful bridge to more advanced logic texts.
Overall, a book which requires more then one read to scratch the surface, and a lifetime to understand it' ...more
Russell’s famous epigram: "pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true."
The ground breaking paper Incompleteness by Godel rocked the very foundation of mathematics and set the edifices of Principia Mathematica (by mathematical greats Bertrand Russel and Alfred Whitehead) crumbling. Understanding what he did and how he did it is the subject of Nagel and Newman's book. Godel's Proof takes the reader stepwise into the idea ...more
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“Like Molière’s M. Jourdain, who spoke prose all his life without knowing it, mathematicians have been reasoning for at least two millennia without being aware of all the principles underlying what they were doing. The real nature of the tools of their craft has become evident only within recent times A renaissance of logical studies in modern times begins with the publication in 1847 of George Boole’s 'The Mathematical Analysis of Logic'.”
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“Contrary to all prior belief, the vast continent of arithmetical truth cannot be brought into systematic order by laying down for once and for all a fixed set of axioms and rules of inference from which every true arithmetical statement can be formally derived.”
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