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Gödel's Proof
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In 1931 Kurt Gödel 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. Gödel 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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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
If you're into stuff like this, you can read the full review.
Higher Order Languages: "Gödel's Proof" by Ernest Nagel, James Newman
What Gödel's Theorem really says is this: In a sufficiently rich FORMAL SYSTEM, which is strong enough to express/define arithmetic in it, there will always be correctly built sentences which will not be provable from the axioms. That, of course, means their contradictions will not be provable, either. So, in a word, the sentences, even though correctly built, will be ...more
Higher Order Languages: "Gödel's Proof" by Ernest Nagel, James Newman
What Gödel's Theorem really says is this: In a sufficiently rich FORMAL SYSTEM, which is strong enough to express/define arithmetic in it, there will always be correctly built sentences which will not be provable from the axioms. That, of course, means their contradictions will not be provable, either. So, in a word, the sentences, even though correctly built, will be ...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
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
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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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
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
This book is great in showing the essence of Gödel’s proof.where regardless of the axiomatic system enlargement a true arithmetic statements exist that cannot be deducible from them. Making the axiomatic system unable in exhausting all mathematical true statements or showing that there are endless true independent statements.Hence rendering the axiomatic system if consistent is also incomplete.
Which also explain Gödel’s ‘platonic realism’ in which mathematical truths are objects we happen to stu ...more
Which also explain Gödel’s ‘platonic realism’ in which mathematical truths are objects we happen to stu ...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.
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This book makes you feel smart for understanding (a dumbed-down version of) the proof of Gödel's incompleteness theorem.
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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
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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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
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...
Very clear. I've wanted to read this for years and a recent paper in data science (which is my field of work) linked the no free lunch theorem to Godel's theorem. After reading this book, I'm not remotely surprised by the link; in fact, it wouldn't surprise me if the author's had finished reading this book themselves and _then_ wrote the paper!
Highly recommend for intense logic formalism-heavy fun. ...more
Highly recommend for intense logic formalism-heavy fun. ...more
My favorite quote from this book is the first sentence of the book's central chapter covering Gödel's famous paper:
"Gödel's paper is difficult."
After reading this book, and the masterwork Gödel, Escher, Bach: An Eternal Golden Braid, I can honestly say that my mind refuses to accept the ideas of Gödel's paper, let alone understand what his results imply.
Gödel, at the age of 25, was able to prove that any consistent formal system able to model basic arithmetic is incomplete. In other words, the ...more
"Gödel's paper is difficult."
After reading this book, and the masterwork Gödel, Escher, Bach: An Eternal Golden Braid, I can honestly say that my mind refuses to accept the ideas of Gödel's paper, let alone understand what his results imply.
Gödel, at the age of 25, was able to prove that any consistent formal system able to model basic arithmetic is incomplete. In other words, the ...more
A fun, concise text that illustrates one of the most profound theorems of mathematics and philosophical logic. Godel's Incompleteness Theorems are some of the most mind-boggling creations of human thought that have profound implications for more human thought. "Godel's Proof" is excellent in that the author provides a relevant background to the proof and its implications, starting with a buildup on the concepts of consistency and meta-mathematics, into the arithmetization of meta-mathematics (
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I had a bit of difficulties to find my way into the book. It seemed verbose and informal in the beginning and the terminology took some getting used to. Also, sometimes the authors overexplain some basics while not really defining/introducing other things properly that were more important in my opinion. I found the elaboration of Russel's antinomy related to defining sets of sets not containing themselves particularly annoying. It is much better and less verbosely described elsewhere.
But the bo ...more
But the bo ...more
A slim, manageable read, and helped boost my understanding of Godel's theorem, though I'm still far from clarity. A good candidate to re-read in the future.
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topics | posts | views | last activity | |
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Science Book Club: Gödel's Proof | 3 | 20 | Mar 27, 2019 06:58AM | |
Goodreads Librari...: Wrong number of pages for Nagel and Newman's Godel's Proof | 5 | 19 | 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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