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Gödel's Proof

4.16  ·  Rating details ·  4,484 ratings  ·  230 reviews
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 the natural ...more
Hardcover, 129 pages
Published October 1st 2001 by New York University Press (first published 1958)
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WarpDrive
Oct 09, 2017 rated it it was amazing
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 com
...more
Riku Sayuj
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 destro
...more
Manuel Antão
Apr 30, 2019 rated it it was ok
Shelves: 2019
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
...more
Matt
May 25, 2009 rated it really liked it
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
...more
Szplug
Feb 20, 2011 rated it really liked it
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 Gödel Numbers were mapp ...more
Sam Ritchie
Nov 11, 2015 rated it it was amazing
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.
...more
Ali Reda
Sep 02, 2014 rated it it was amazing  ·  review of another edition
Shelves: philosophy
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
...more
David Olmsted
May 04, 2012 rated it it was amazing
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 ...more
Jodh Singh
Apr 24, 2017 rated it it was amazing
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.
Joshua Nomen-Mutatio
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
Chayan Ghosh
Oct 30, 2016 rated it really liked it
Shelves: mathematics
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
...more
Adam
Oct 18, 2011 rated it really liked it
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.
Jafar
Aug 07, 2017 rated it really liked it
This book makes you feel smart for understanding (a dumbed-down version of) the proof of Gödel's incompleteness theorem.
Mahdi Dibaiee
Feb 28, 2017 rated it it was amazing
A fun and thought provoking read indeed, would recommend it to anyone who
* loves paradoxical statements
* would like to know more about mathematical logic
Mengsen Zhang
Jan 20, 2013 rated it really liked it
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 ...more
Nathan Glenn
Apr 29, 2013 rated it really liked it
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 d
...more
Bob Finch
May 18, 2013 rated it really liked it
Recommends it for: geeks
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 ...more
Ed
Aug 03, 2011 rated it really liked it
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 ...more
Aysja Johnson
Jul 16, 2017 rated it it was amazing
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
Sherwin
Aug 12, 2007 added it
Recommends it for: everybody
Shelves: philosophy
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...
Gabe Koss
Sep 09, 2017 rated it really liked it
Shelves: philosphy
This was an unexpectedly quick read. I dove right in an found it to be quite rewarding and moderately accessible. It does a great about framing Gödels Incompleteness Theorem but also introduced me to a lot of the history of logic and many important concepts.
KarLuis
Sep 06, 2016 rated it it was amazing  ·  review of another edition
A gently accessible and highly readable exegesis of what I feel is the most difficult text a student of philosophy can attempt to read. (Yes, much more so than Kant.)
Darian
Jul 10, 2019 rated it it was amazing
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 ( ...more
Gary
Feb 14, 2018 rated it it was amazing
Learning a new language. Fascinating.
Ben
Aug 12, 2018 rated it really liked it
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.
Crito
May 19, 2018 rated it really liked it
Shelves: nonfiction
Very accessible for mathlets such as myself.
Mihir
Mar 07, 2018 rated it it was amazing  ·  review of another edition
Delightful read, dumbed down perfectly to my comfort level. You can see glimpses of what inspired GEB as you go through the book and marvel at how Douglas Hofstadter was able to build on those ideas.
Lara Thompson
Mar 11, 2019 rated it it was amazing
Shelves: technical
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.
D Schmüdde
May 26, 2018 rated it really liked it
One of my favorite aspects of this book is the co-author's willingness to reiterate certain concepts. For example, it is stated several times that Gödel's results could not be resolved by expanding the axiomatic system; each time the Nagel and Newman took the opportunity to shed further light on this conclusion within a new context. I imagine there are nuggets in here for more advanced readers, or they can move through this text quickly. Either way, it was extremely helpful to me.
Kirill
Jan 01, 2012 rated it it was ok  ·  review of another edition
'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 ...more
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Science Book Club: Gödel's Proof 3 18 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'.” 3 likes
“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.” 2 likes
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