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Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries)

3.95  ·  Rating Details ·  984 Ratings  ·  91 Reviews
KURT GODEL IS CONSIDERED the twentieth century's greatest mathematician. His monumental theorem of incompleteness overturned the prevailing conviction that the only true statements in math were those that could be proved. Inspired by Plato's philosophy of a higher reality, Godel demonstrated conclusively that there are in every formal system undeniably true statements that ...more
Paperback, 224 pages
Published February 28th 2006 by W. W. Norton & Company (first published 2005)
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Jan 21, 2011 Szplug rated it really liked it
The more I think about language, the more it amazes me that people ever understand each other at all.

Fucking Gödel.

The above (pictured with a rueful smile and head shake) succinctly summarizes my feelings for the incomparable Kurt Gödel—the greatest logician since Aristotle, as Rebecca Goldstein makes sure to iterate several times—the quiet and unassuming genius whose steel-trap mind could capture those ethereal abstract truths and convert them into human language constructs; who single-handedly
Kyle York
Mar 20, 2013 Kyle York rated it it was amazing
What a wonderful book. Goldstein not only lays out Godel's famous theorems in relatively understandable terms for the layman (an accomplishment in itself,) but provides an original, funny, and lucid account of the intellectual atmosphere in which these theorems arose. She discussed Godel's relation to the Logical Positivists and Formalists, which sheds great light upon the meaning of his discoveries. She also dispels the postmodernists mythologies about what Godel's theorems mean. In addition, s ...more
Brad Lyerla
Nov 18, 2013 Brad Lyerla rated it really liked it  ·  review of another edition
INCOMPLETENESS is an excellent book about an intellectually elusive subject. Kurt Godel's fame was established by his proof of something called "the Incompleteness Theorem." His proof employed formal logic to establish a basic truth about mathematics. Namely, that in closed systems, there will be true statements that cannot be proved. Until Godel's proof, many leading mathematicians assumed the opposite was true. This is a challenging subject to write about, but Goldstein makes it easily accessi ...more
Keith Akers
May 21, 2010 Keith Akers rated it really liked it
This is a great book to learn more about "Goedel's Proof" (actually two proofs, or actually three proofs if you count his Ph. D. thesis on predicate calculus). The incompleteness of mathematics is an astounding concept -- it's so astounding, that you are left breathless, not even sure what the whole thing means. Does this mean that God exists? Actually, Goedel himself toyed with variations of the ontological proof. The incompleteness of mathematics is just really hard to wrap your brain around; ...more
Apr 24, 2016 Domhnall rated it it was amazing
Shelves: philosophy
This book is succinct, accessible and well constructed. Godel's Incompleteness Theorems are so significant in the history of ideas that it is essential to have a decent grasp of just what they are and why they mattered and this book supplies that need for general readers. It gives a good enough explanation of Godel's findings and deals with the reactions of other major names to his theories, which sheds interesting light on their work too.

We need to grasp Godel's theories accurately because we
Collin Winter
"Incompleteness" is less about Gödel's actual incompleteness theorems -- the proofs and their specific mathematical legacy -- than it is about the philosophical environment those theorems were developed in. Put another way, this is a book less about Gödel and more about Gödel and Wittgenstein, or perhaps more accurately, about Wittgenstein and Gödel.

This is a book that prefers to tell rather than show: Goldstein spends 160 pages telling the reader how amazing and important and revolutionary Göde
Apr 23, 2007 Davin rated it liked it
Well written and a good picture of Godel, his work, philosophy and the times he lived in. There would be more starts up there but for 2 reasons:

1 The book goes through thumbnail sketches of Godel's famous proofs and then a more involved version, but even after the more detailed explanation I still felt like I had only scratched the surface of it. Some of the things asserted about the process of Godel numbering seemed almost magical as a result. This is a tough balancing act for any popular take
Jul 13, 2008 Matthew rated it it was ok
Shelves: real-worldy
Suck it, postmodernists.
Jun 08, 2014 Rossdavidh rated it liked it  ·  review of another edition
Shelves: purple
David Foster Wallace (RIP) once referred to Kurt Goedel, the subject of this book, as mathematics' Prince of Darkness. Douglas Hofstadter gave his Incompleteness Theorems (1 and 2) a central role in his book “Goedel, Escher, Bach”. Goedel's ideas are so central to 20th century thinking that it is likely that Einstein (for many years until his death Goedel's closest friend) was the only person he ever met who he was not, eventually, to become more famous and influential than (for example Wittgens ...more
Apr 16, 2014 Brett rated it liked it  ·  review of another edition
This book is serviceable in being a readable introduction to Gödel's famous proofs and a general (very general) understanding of the intellectual and historical circumstances within which it grew. While I do not have the capabilities to judge whether she provides a completely faithful representation of Gödel and his proofs, it is within my power to deem her handling of Wittgenstein, whom she spends a great deal of pages discussing, to be sorely unsympathetic, uncharitable, and misrepresentative. ...more
Mar 15, 2007 Jafar rated it really liked it
A great book about this most genius mathematical mind of the last 100 years. Rebecca Goldstein actually provides a simplified version of the proof of the Incompleteness Theorem.
Feb 14, 2016 Micah rated it it was amazing
This book is perfectly balanced between the historical and analytical. The last line in the book was worth all the effort and shoots the book up several layers of appreciation.
Jan 14, 2017 Daniel rated it really liked it
This book is an excellent overview of the life and work of the greatest mathematician of the twentieth century. I love the way the book ties the biographical, philosophical, and mathematical all together -- describing how Goedel's philosophical worldview drove him to search for an absolute proof of his point of view. The treatment of the famous incompleteness theorem seems to me well done, though I confess I haven't studied mathematical logic in over 20 years and pretend to no deep understanding ...more
Alex Lee
Mar 10, 2016 Alex Lee rated it really liked it
Over the last two decades two I've read about Godel off and on. So I am familiar with the logic but not with the man. Goldstein is able to present both in this tiny book in a way that is compelling and thorough. She talks about the man but mainly about the evolution of his ideas. The main piece of information she gives me that is different is that Godel is a Platonist, which makes a great deal of sense. Seen from this way, the Incompleteness Theorem can be understood as a simple translation erro ...more
Jul 21, 2015 Nick rated it liked it
Shelves: math
Godel's incompleteness theorem proves that, in any consistent formal mathematical system sufficiently complex to be useful (for instance, the one we use, with our pluses and minuses and our ten digits), there must exist certain arithmetical statements which are true, but impossible to prove. The gist is that it's impossible to remove all intuitive statements--for instance, that a=a--from mathematics, without rendering our system inconsistent, i.e. broken. This was, and still is, a huge deal for ...more
Omar Rodriguez-Rodriguez
Oct 02, 2016 Omar Rodriguez-Rodriguez rated it really liked it
Half bio, half review of Godel's proof. It is a good overview of his life, and a peek inside his mind, interests and late paranoias.
May 21, 2010 Lynn rated it really liked it
This added to my understanding of philosophy more than mathematics, but the book was primarily a biography of Gödel, so the fact that a reader can actually take away some philosophical insights from a biography makes this book unique.

The description of Gödel’s proof was short and well written, but it was only about 25 pages. I can say I have less of a misunderstanding of the proof, but I can’t say I understood Goldstein’s explanation. If you’re looking for a good explanation, this probably isn’t
Kirk Lowery
This book is an intellectual biography of one of the greatest mathematicians and logicians of the 20th century. Gödel single-handedly derailed the Hilbert-Russell program of putting mathematical proof on to a purely mechanistic (formal) basis. He showed that in any formal system that includes arithmetic, there will be true propositions that cannot be proved within that formal system. There is much more to it than that, but the primary implication is that mathematical truth exists apart from how ...more
Jamie Collins
Probably this is a really good book, but it’s intended for a more erudite reader. It was largely over my head. It's part of the Norton "Great Discoveries" series, and the other two books in that series I've read (on Marie Curie and Ignac Semmelweis) were more accessible than this one.

The introduction is nice - she talks about the creation of the Institute for Advanced study in Princeton, and the unlikely friendship between Gödel and Einstein. Then she launches into a discussion of the Vienna Cir
Mar 16, 2011 Brian rated it liked it  ·  review of another edition
I found half quite interesting

Perhaps it's narrow-minded of me, but I didn't really care too much about Gödel's upbringing and his (non-proof-based) philosophical views. It is good to cover early history in a biography, but the focus should have been on his incompleteness work. Goldstein spends far too much time on his philosophical views (to the point of feeling quite redundant to me) and how they contrasted with other leading thinkers of the time. I found her focus on this topic inexplicable (
Mike Tracy
Sep 10, 2015 Mike Tracy rated it really liked it
This was a very tough read for me, especially the middle section, where the author gets into a lot of discussion of Godel's layering and numbering system of proof. Very abstract and very difficult for me to follow, but, in the more general sections, an enjoyable, if still somewhat challenging a read. If I'm getting the central idea of the book, it deals with Godel's revolutionary idea that truth or falsity cannot be derived by reasoning from any closed set. Inevitably such a procedure will alway ...more
Mar 27, 2010 Ken rated it really liked it  ·  review of another edition
The mind of a mathematician must be a horrible place to be. It must also be a difficult thing to dissect, but Rebecca Goldstein did so fabulously. As is my preference for reading, I picked this book up for the biographical information, yet found myself studying the technical breakdown of the incompleteness theory. I don't know how well it set in my mind, but my curiosity was piqued. I will definitely look into it more.

One thing that bothered me a little, and I read this in another review as wel
Peter Ibsen
It's interesting that this website doesn't have an option for "read most of but gave up more then halfway through".
I had been planning on reading this for a while. Some years ago I was very interested in physics, mathematics and the great minds that formed the foundations of what I studied. "Incompleteness" was recommended to me by another student and I finally have gotten around to starting it, getting halfway through and then putting it down for good. It is not a bad book by any means. I just
Jul 03, 2014 Mary rated it really liked it
Overall, a very engaging read. For the mathematically illiterate,Ike myself, all but one of the chapters is fully comprehensible and the confusing chapter filled with formulas is not necessary to understand and enjoy the overall book and its themes. Goldstein tells us of Godels life and his incompleteness theories' effects on math and philosophy. Perhaps the most intriguing issue she raises is to what extent can his theories be used to understand the "mind"? If formal systems do have an Undecida ...more
Sahil Moza
Aug 28, 2015 Sahil Moza rated it really liked it
Recommended to Sahil by: Vishaka Datta
Well written book about Godel's life. Has a dedicated section for the incompleteness theorems, but it could have been explained better, and is still for a reasonably mathematical audience. So if you are looking for incompleteness theorems explained in layman terms, this book is probably not for you. But if you want to know more about Godel's life and times at the Vienna Circle, the rise and (kind of) fall of logical positivism, and Godel's later life at Princeton, and his friendship with Einstei ...more
Jan 19, 2016 Jake rated it really liked it
Much of Gödel's work is the basis for much of modern proofs in mathematics and this books is a focus on his incompleteness theorem which in affect the keystone of his contributions to this discipline. It states that essentially in any closed arithmatic system there are true statements that cannot be proved. For instance, A=B/C=>C=B/A or A=--A. Both are true but cannot be proven. This may not seem like a big deal but in Gödel's day most mathematicians believed the opposite. I don't really know ...more
Scarlett Sims
When I first started reading this book I thought it was basically going to be a biography of Godel, but it's much better than that. It discusses his proofs, but also why they were so important, especially in the context of other scientific, mathematical, and philosophical thought of the time. The author does write a little bit about his personal life, mostly his later years, which is also quite interesting. I really wish, however, that she had gone into more detail about the actual content of hi ...more
Oct 19, 2007 elizabeth rated it really liked it
I've read Goldstein's fiction and often been a little turned off by her academic snobbery. But this book, despite its exuberant praise of notions of genius, got me excited about logic in a way I hadn't felt since undergraduate days. And lest you thought the life of the logician was all structure, Goldstein lets us know that Godel enjoyed walks with Einstein, reading Kafka and watching Disney's the Seven Dwarves.
Perhaps it's Godel's quirkiness (which dissolves in later life to more profound menta
Aug 18, 2015 Bryan rated it really liked it
Short but fascinating book on the greatest mathematical result of the 20th century. For the most part, Goldstein makes it accessible to the layman, although I did find I had to Google certain mathematical terms I had forgotten. Also, the simplified derivation of the result sometimes caused me to pause and reread certain passages until I understood them. If you enjoy and can follow pop science books such as 'A Brief History of Time,' this book shouldn't present a problem. Although, Goldstein does ...more
Jun 17, 2010 Rod rated it it was amazing
An important and accessible treatment of the Godel incompleteness theorems, perhaps the most important mathematical development of the 20th Century.

The incompleteness theorems combine as a statement in formal number theory that there exist undecidable formulae, that can be neither proven or disproven. In simple terms, this means there are formulas that are true, but cannot, even in theory, be proven true. The implications of the theorems are profound, both in mathematics and epistemology.

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Rebecca Newberger Goldstein grew up in White Plains, New York, and graduated summa cum laude from Barnard College, receiving the Montague Prize for Excellence in Philosophy, and immediately went on to graduate work at Princeton University, receiving her Ph.D. in philosophy. While in graduate school she was awarded a National Science Foundation Fellowship and a Whiting Foundation Fellowship.

After e
More about Rebecca Goldstein...

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“The necessary incompleteness of even our formal systems of thought demonstrates that there is no nonshifting foundation on which any system rests. All truths — even those that had seemed so certain as to be immune to the very possibility of revision — are essentially manufactured. Indeed the very notion of the objectively true is a socially constructed myth. Our knowing minds are not embedded in truth. Rather the entire notion of truth is embedded in our minds, which are themselves the unwitting lackeys of organizational forms of influence.” 5 likes
“The secret of the demagogue is to appear as dumb as his audience so that these people can believe themselves as smart as he is.” 2 likes
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