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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 in
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Hardcover, 129 pages
Published
October 1st 2001
by New York University Press
(first published 1958)
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(showing 1-30)
Highly entertaining and thoroughly compelling, this little gem represents a semi-technical but comprehensive and mathematically accurate elucidation of the famous (and so often misused and misunderstood) Godel's meta-mathematical results concerning the limits of provability in formal axiomatic theories.
Being relatively short, this book does not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing (in theory of computability, ...more
Being relatively short, this book does not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing (in theory of computability, ...more
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
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
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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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
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
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
Untuk sebuah karya (pemudah) matematik, buku ini sebenarnya sangat mudah untuk dibaca; lebih mudah daripada apa yang aku bayangkan. Aku fikir, ada dua sebab: (1) penulis berjaya memberikan gambaran jelas tentang apa yang Godel telah usahakan, dan penulis telah abaikan sejumlah pembuktian formula yang ia rasakan tak perlu; dan (2) aku membaca buku ini dengan maksud meninjau apa yang Godel katakan, dan bukan kenapa apa yang Godel katakan itu benar. Jadinya, sepanjang pembacaan, aku tidak begitu te
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Other reviews here do an excellent job of going over the book's subject matter. I'd encountered the proof before in "Gödel, Escher, Bach" - this book gives a really deep, clear dive into the main thrust of Gödel's proof that it's not possible to write an absolute proof of consistency for a formal system that can describe number theory inside that theory itself. (The book will teach you what everything in that phrase means, so don't be scared!)
Here's my more personal review.
I'm a functional progr ...more
Here's my more personal review.
I'm a functional progr ...more
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
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 first 80 pages are a breeze, but pages 80-92 need many re-reads.
P.S. Remember to not miss-use the incompleteness proof to give sweeping and profound statements about nature of the world or other mumbo jumbo. Godel went to a great deal of trouble to write a very exact and general proof, do not desecrate his work to prove weird stuff.
P.S. Remember to not miss-use the incompleteness proof to give sweeping and profound statements about nature of the world or other mumbo jumbo. Godel went to a great deal of trouble to write a very exact and general proof, do not desecrate his work to prove weird stuff.
How do I come up with a fair review for this book, without having my judgement clouded by the genius of Godel? On one hand I am speechless by the ingenuity of the proof devised by Godel and what it signifies, while on the other I am disappointed with the authors for how insufficiently the legend's mind has been probed and represented in these pages. For a book named Godel's Proof, this one barely scratches the surface.
On the plus side, it was a very involved and difficult topic, and it was a bol ...more
On the plus side, it was a very involved and difficult topic, and it was a bol ...more
I have pondered Gödel's famous proof since first hearing of it, but I could never manage the original paper. This compact, enlightening little gem shines a bright light on that remarkable piece of work, and on logical thinking in general. In a stunning surprise to the the status quo, Gödel shook the foundations of mathematics by demonstrating it to be incomplete or inconsistent. Hofstadter's edition with his excellent forward is particularly helpful for understanding the continued impact of Göde
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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...
Oct 29, 2016
Ege Özmeral
rated it
really liked it
·
review of another edition
Shelves:
mathematics,
philosophy-of-mathematics
Amazing book with deep and sometimes complex concepts , I really enjoyed reading it.
I skipped some parts but I think I understand the subject better after reading It.
Here is what I learned from Godel's proof:
Incompleteness theorem.
1) Mathematical systems that we know (like Geometry) are called axiomatic systems , We accept some things as axioms and derive other facts (theorems) from that set of axioms .
e.x : if a=b and b=c then a=c
Now Godel Showed that , "No matter what our set of axioms are , ...more
I skipped some parts but I think I understand the subject better after reading It.
Here is what I learned from Godel's proof:
Incompleteness theorem.
1) Mathematical systems that we know (like Geometry) are called axiomatic systems , We accept some things as axioms and derive other facts (theorems) from that set of axioms .
e.x : if a=b and b=c then a=c
Now Godel Showed that , "No matter what our set of axioms are , ...more
I've encountered Gödel's proof(s) in various settings -- once in a formal logic course and once in a philosophy of mathematics course. Both times I was too far in the weeds to really glean the huge importance of his work. This book does incredible justice to the intellectual masterpiece Gödel constructed, and it does so in a very incisive way. It also places his work in the context of the mathematic research at the time which gives it much more meaning.
I found this book fairly easy to read with ...more
I found this book fairly easy to read with ...more
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
Dec 27, 2011
Maurizio Codogno
rated it
really liked it
·
review of another edition
Shelves:
matematica,
logica
Il teorema di incompletezza di Gödel è uno dei temi più ricorrenti nella divulgazione matematica, solo che non è mai facile riuscire a trovare una spiegazione semplice. Questo breve saggio probabilmente è stato il primo a portare la dimostrazione al grande pubblico - grande si fa per dire, visto che comunque occorre tutta una serie di conoscenze di base. La dimostrazione è preceduta da un excursus che spiega non solo la storia di come la matematica abbia seguito una corrente formalista nata alla
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'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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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
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Goodreads Librari...: Wrong number of pages for Nagel and Newman's Godel's Proof | 5 | 18 | Mar 31, 2015 07:01AM |
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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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Oct 17, 2017 07:58AM
updated Oct 17, 2017 12:53PM