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Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries) Paperback – 28 Feb 2006

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Product details

  • Paperback: 224 pages
  • Publisher: W. W. Norton & Company; Reprint edition (28 Feb 2006)
  • Language: English
  • ISBN-10: 0393327604
  • ISBN-13: 978-0393327601
  • Product Dimensions: 1.4 x 0.2 x 2 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: 90,416 in Books (See Top 100 in Books)

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Product Description


"Magnificent.... A stimulating exploration of both the power and the limitations of the human intellect.... Goldstein is an excellent choice for this installment of Norton's Great Discoveries series: Her philosophical background makes her a sure guide to the underlying ideas, and she brings a novelistic depth of character and atmosphere ... to her sympathetic depiction of the logician's tortured psyche, as his relentless search for logical patterns ... gradually darkened into paranoia."

About the Author

Rebecca Goldstein is a MacArthur Fellow, a professor of philosophy, and the author of five novels and a collection of short stories. She lives in Cambridge, Massachusetts.

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8 of 8 people found the following review helpful By Simon Pollack on 15 Sep 2009
Format: Paperback
Rebecca Goldstein's book can be described as epic in its approach to the theory of knowledge. And, as a philosophy layman (though I'm mathematically literate), it is definitely understandable while deep enough to require several re-reads of previous sections to follow the story.

It is, in fact, a story. It has a central theme, appropriately reaching its technical summit in the middle of the book: namely Goedel's famous theorem(s). It provides a wonderful discussion of the context historically (the characters, the places, the intellectual developments) of Goedel's work. And it provides a very accessible account of the modern history (from late 19th century) of the relevant branches of philosophy. Notably, it covers how Goedel's work was inspired by, and then effectively defined, what it is that can be said about "mathematical truth" (apologies for the slackness of this phrase's meaning: read the book and you will know what I mean).

Highly recommended to any interested readers willing to put in a little effort to understand a challenging subject.
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2 of 2 people found the following review helpful By Gregory Connor on 21 Jan 2013
Format: Hardcover Verified Purchase
I was familiar with Godel but knew little about Wittgenstein before reading this short but enchanting book about their mid-20th century clash of ideas. Two very interesting characters-- so different but each brilliant in his own way. This book also provides clear insights into Godel's key contributions to mathematical logic, for the layman like myself.
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5 of 6 people found the following review helpful By Antonios F. Arkas on 22 Feb 2010
Format: Paperback
This book primarily constitutes Gödel's biography and an analysis of the historical background from which the proof immerged and secondary an outline of the proof. The historical presentation is successfully entangled with the very intricate and interesting epistemological and philosophical ramifications of this great intellectual achievement. Finally the book is supplemented with an attempt to explain the main features of the proof. I concider the first three goals i.e. biography, historical context and analysis of proof's ramifications to meet with success but not so for the last one concerning the proof outline.

Every scientific idea - especially those which were so revolutionary that managed to shake the foundations of science and knowledge - seem naked, purposeless and unintelligible when presented without the accompanying historical context. Since history is created from special charismatic and dramatic persons, a historical approach inevitably comprise a biography and an account of the ideas which influenced the revolutionist thought. Ideas could be thought as intellectual species which evolve in human minds under the pressure of reality and the discovery of new inexplicable facts. Rebecca Goldstein successfully manages to achieve these two goals with an excellent presentation of the persons and the ideas relevant to Gödel's proof. Her narrating style makes these intricate epistemological and philosophical ideas much more easier to digest and pleasant to read.

One of the aspects which makes this book an excellent choice for reading is it's integrity.
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Most Helpful Customer Reviews on (beta) 79 reviews
96 of 106 people found the following review helpful
A fascinating subject 6 Jun 2005
By bensmomma - Published on
Format: Hardcover
Although I'll bet that readers more versed in the history of mathematics and philosophy will wish for more than Goldstein offers, I found "Incompleteness" to be a fascinating and well-written introduction to both Godel and the philosophy behind his incompleteness theorem (which proves, mathematically, that in any formal system, such as arithmetic, there will be propositions that are unprovable even though true).

Goldstein is such a clear writer that I finished the book feeling I actually understood this logic. More than simple clarity, though, she conveys a genuine affection for the subject (both Godel and his proofs). You can feel why she gets all worked up about its philosophical implications. It doesn't feel obscure in the least. How much writing about philosophy can say as much?

If you are looking for a complete description of ALL Godel's life work, you won't be happy (she deals almost exclusively with the incompleteness results, not his other work). Nor will you find this to be a standard-issue narrative biography (birth, education, marriage, death); although you can extract the basic facts from Goldstein's scant 260 pages, Godel's wife Adela doesn't appear until page 223; Godels' difficulties with his mental health are treated as non-issues rather than as defining or formative events.

In the end, it's all about the math, and I enjoyed it.
166 of 187 people found the following review helpful
Good, not great 11 Mar 2005
By C. Bracken - Published on
Format: Hardcover
I was not as enthusiastic about this book as most other reviewers seem to be. This is a book with some important high points, but also some serious flaws.

I was disappointed mainly by the biographical parts of the book. This is a very dry retelling of what is known of Godel's life. Other biographies of seemingly boring mathematicians have been engrossing (read the excellent "The Man Who Knew Infinity" or "A Beautiful Mind"), but this book misses the mark in terms of giving us a picture of who Godel really was.

Godel was part of the Vienna Circle, so we get a lot of history about the Vienna Circle in general. He was later at the Institute for Advanced Studies, so we get a lot of IAS history. But we seem to get little about the man himself, and more about the groups around him.

The meat of the book focuses on the Vienna Circle, and the author's main point: that Godel was a Platonist among the Positivists, and that his incompleteness theorems have been hijacked and misinterpreted by positivists over the years. This part is important and interesting, but I would have liked to have heard more about Von Neumann (who gets only a brief mention) and less about Wittgenstein. But this is in keeping with the book's bias toward the philosophical side of the story.

The explanation of Godel's main proof seemed a bit unclear to me, but I give Goldstein a lot of credit for not simply glossing over the details like most over-simplified explanations one reads. My suspicion is that most who read this book's description of the proofs will laud its clarity while quietly admitting to themselves that they didn't quite understand it all.

Bottom line: if this book will be on your bookshelf next to books of philosophy and logic, it will make a welcome addition. If you are setting it next to "Men of Mathematics," "The Man Who Knew Infinity," or other such rich biographies of mathematicians, you may be disappointed by this book's philosophical and academic tone.
50 of 55 people found the following review helpful
2 books on Kurt Gödel; the authors should have collaborated 2 Jun 2005
By Jesse Steven Hargrave - Published on
Format: Hardcover
It seems to me that, with increasing frequency, two books on the same or closely related subjects come out from different publishers almost simultaneously. I suspect an epidemic of corporate espionage. In 2003/4, did we really need two books with the identical title "Lincoln at Copper Union" about a pre-campaign speech in New York by the eventual president? Why was "The Empire of Tea" published within 6 months of "Tea: Addiction, Exploitation, and Empire"? (Perhaps they were tied to an epic mini-series that I missed.)

Kurt Gödel and his work have been largely ignored of late, yet now we suddenly have two books attempting to resurrect interest. Palle Yourgrau's "A World Without Time: The Forgotten Legacy of Gödel And Einstein" was published in January 2005, and "Incompleteness: The Proof and Paradox of Kurt Gödel" by Rebecca Goldstein just one month later.

Both are small-format books, and thus both attempt to squeeze already dense subject matter into unreasonably constricted space. Both use Gödel's personal and intellectual friendship with Einstein as a systematizing motif. Each author dedicates considerable time to rehearsing the history of The Vienna Circle, where Gödel spent formative years, and the Institute for Advanced Studies in Princeton, where Gödel and Einstein completed their careers. And both Goldstein (a mathematician and novelist) and Yourgrau (a professor of philosophy) attempt to give a summary of Gödel's important theorems that would make them accessible to the non-specialist.

However, the two books differ in important respects.

Goldstein, when dealing with Gödel's professional work, focuses almost exclusively on that concerned most directly with mathematical logic: his Incompleteness Theorems. That means Gödel's more cosmological exertions, which came after he joined the Institute, are left untreated. And Goldstein has a theorem or two of her own: that the implications of Gödel's work in mathematical logic and metaphysics were seriously misconstrued even in his own day, that such misunderstanding was a gnawing disturbance to the logician, and that it contributed greatly to his increasingly pathological alienation from his colleagues and the world at large.

Yourgrau is more interested in the validity and implications of Gödel's later philosophical (or cosmological) work on the nature of time. Yourgrau published an earlier monograph which the book jacket claims "sparked a resurgence of interest in Gödel's ideas about time and relativity." Yourgrau comes across as Gödel's self-appointed apologist, armed to defend the logician against claims that these later philosophical applications were amateurish and easily dismissed.

Both books, I felt, succeeded in gaining the reader's sympathy for their respective perspectives. But neither could be suitably comprehensive in the relatively few pages allotted them. For me, Goldstein did the slightly better job of explaining the Incompleteness Theorems.

(It would be beyond the skills of even the most accomplished popularizer to fit a truly satisfying explanation into these abbreviated books. The reader is subjected in both to sentences such as this one from Yourgrau: "The representation occurs via the arithmetization of the syntax of FA, so corresponding to a given syntactical truth Bew(x,y) of MFA, there is an arithmetical truth Bew(x,y) of IA that corresponds to a formula Bew(x,y) in FA that can be interpreted as saying that the sequence of formulas with Gödel number x is a proof of the formula with Gödel number y, and this formula, Bew(x,y), is a theorem of FA.")

You thus get from Goldstein a better grounding in what is considered Gödel's true legacy. But you have to look to Yourgrau to get even a basic sense of what Gödel later had to say about cosmology. In that sense, Yourgrau's book is the more thought-provoking.

Both authors are gifted writers, although Yourgrau seems to loose some control over his metaphors as he gets increasingly worked up about the lack of respect given to Gödel's cosmological contributions. As Yourgrau tells of a 1995 symposium on "Gödel's General Philosophical Significance", readers may feel they have stumbled into a metaphysical food-fight.

The fact that these two books were published at almost the same time shows that there must be a significant audience of non-specialist readers interested in an updated accounting of Gödel's life and work. It's unfortunate that such readers have to buy both these books and navigate through so much redundant material to get even the beginnings of a complete perspective.
26 of 30 people found the following review helpful
Poor explication of theorem; too much Wittgenstein 10 July 2006
By Edward T. Brading - Published on
Format: Paperback
The book has two main problems. The first is Goldstein's explication of the incompleteness theorem. The theorem is the reason for reading a book about Goedel. For the most part, the worth of a book about him for a general reader is measured by the clarity of an explication of the theorem. Goldstein's audience comprises readers who are not logicians or mathematicians, and so a lack of rigor is expected (p. 172). But Goldstein simplifies too much. Her explication is somewhat less clear than both the longer explication in Goedel's Proof by Nagel and Newman and the more technical introduction by Braithwaite in the Dover Publications reprint of Goedel's original paper.

Goldstein's numbering system (p. 172-175) is an example of oversimplification. Goedel's numbering system "used the exponential products of prime numbers and relied on the prime factorization theorem which states that every number can be uniquely factored into the products of primes" (p. 172). In this way, the "metasyntactic relation of provability will become an arithmetical relationship" (p. 176). Under Goldstein's simplified system of numbering, however, it is not at all clear that provability relationships among propositions would translate into arithmetical relationships among numbers, as they do under Goedel's numbering system. Why does Goldstein offer an alternative numbering system for illustrative purposes? I can't tell. She says that her system, if it were made rigorous, would be just as complicated as Goedel's own system (p. 172). So rather than invent her own, why doesn't she just set out a non-rigorous version of Goedel's own system? (That is what Nagel and Newman do.) Not only does Goldstein not improve matters, but also she loses clarity. In her illustration, for example, she says: "Suppose that GN(wffsub1) = 195589 and GN(wffsub2) = 317" (p. 175). But under Goldstein's own numbering system, 195589 and 317 would correspond, respectively, to ~'00)' and x~(, neither of which is a wff! By oversimplifying, Goldstein has made a mess.

The second main problem with the book is Goldstein's fascination with Wittgenstein and her comparison of him with Goedel. Any comparison between the two thinkers feels strained to begin with, and Goldstein's book does nothing to allay that feeling. It is a bit like writing a book about Vladimir Horowitz and then devoting a third of the book to comparing him with Liberace. Moreover, apart from whether any comparison is useful, Goldstein refuses to take Goedel at his word when he says that Wittgenstein had no influence on his work (p. 115-116). In fact, Goldstein takes Goedel's emphatic denial, coupled with what she sees as Goedel's resentment of Wittgenstein (p. 89), as evidence that Wittgenstein must have had an influence on Goedel, or "incentive" or "significant, if ambiguous, role," as Goldstein puts it (pp. 89, 116). This is just weird. If Goedel had written that he hated rock candy and didn't even like the looks of it, would Goldstein conclude either that Goedel really did like rock candy or that he ate filet mignon as a substitute? The ink that she spills on Wittgenstein could have been put to better use on Turing or von Neumann, both of whom get too few words.

The book contains strange repetitions. For example, twice Goedel's work is compared to Alice in Wonderland (p. 170, 252) and twice Goldstein tells us that her New York apartment has only one bathroom (p. 140, 184). Her catty remarks about Goedel's wife, his diet, and their home decorating are rude and irrelevant (pp. 208-209, 223). How poorly Goedel dealt with faculty politics is dull and irrelevant (pp. 234-245). Some extra proofreading wouldn't have hurt, either: "GN(p)" should be "GN(psub1)" (p. 174); "tilda" should be "tilde" (p. 174); "swiped" should be "swapped" (p. 210); and Waismann's name is misspelled twice (p. 105).

The book has its good points. The stories of Goedel's quirks and his friendship with Einstein are entertaining, the sketch of the Vienna Circle is okay, and Goldstein is right to point out that Einstein and Goedel should not be lumped together with Bohr, Heisenberg, and others as "destroyers of objectivity" (pp. 38-39). But that's about as good as the book gets.
28 of 34 people found the following review helpful
Sorry Wittgenstein -- there ARE surprises in mathematics 10 July 2005
By Scott J. Aaronson - Published on
Format: Hardcover
(Note: I'm reposting this review, since apparently it was deleted due to a software glitch.)

I didn't expect to learn anything from this book, but (as she did with several of her novels) Rebecca Goldstein surprised me. The book is basically a prolonged attempt to get inside Godel's head -- which, as a previous reviewer noted, means that it talks a lot about philosophy. That was fine with me, since I already knew the math and the basic facts of Godel's biography. Goldstein tells what I think is a new but largely persuasive story: (1) that Godel saw his incompleteness results as affirming the Platonic reality of the integers, and their irreducibility to Wittgensteinian language games; (2) that this was part of his motivation for proving the theorems (especially after he saw firsthand the Vienna Circle's unjustified Wittgenstein-worship); (3) that most people interpreted the theorems as showing the exact opposite of what Godel intended; and (4) that this lack of comprehension was one reason for Godel's paranoia and isolation at IAS, particularly after his fellow realist Einstein died.

My one criticism is that, in explaining the incompleteness results themselves, the book follows a needlessly cumbersome "old-school" approach. The only reason Godel had to futz around with prime numbers for 30 pages is that the concept of a computer had not yet been invented! Once you have Turing machines, the proof of the first incompleteness theorem is maybe one sentence ("If arithmetic was complete, then we could solve the halting problem, but we can't").

As a side comment, the book says little about Godel's work on the continuum hypothesis, and nothing at all about his remarkable letter to von Neumann, which first posed the "P versus NP" question. I consider both of these contributions to be on a par philosophically with the incompleteness theorems. But perhaps they're a subject for a different book.
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