Amazon.com
Gödel's incompleteness theorem--which showed that any robust mathematical system contains statements that are true yet unprovable within the system--is an anomaly in 20th-century mathematics. Its conclusions are as strange as they are profound, but, unlike other recent theorems of comparable importance, grasping the main steps of the proof requires little more than high school algebra and a bit of patience. Ernest Nagel and James Newman's original text was one of the first (and best) to bring Gödel's ideas to a mass audience. With brevity and clarity, the volume described the historical context that made Gödel's theorem so paradigm-shattering. Where the first edition fell down, however, was in the guts of the proof itself; the brevity that served so well in defining the problem made their rendering of Gödel's solution so dense as to be nearly indigestible.
This reissuance of Nagel and Newman's classic has been vastly improved by the deft editing of Douglas Hofstadter, a protégé of Nagel's and himself a popularizer of Gödel's work. In the second edition, Hofstadter reworks significant sections of the book, clarifying and correcting here, adding necessary detail there. In the few instances in which his writing diverges from the spirit of the original, it is to emphasize the interplay between formal mathematical deduction and meta-mathematical reasoning--a subject explored in greater depth in Hofstadter's other delightful writings. --Clark Williams-Derry
Book Description
"A little masterpiece of exegesis." -Nature
"An excellent nontechnical account of the substance of Gödel's celebrated paper." -American Mathematical Society
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 sciences-perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."
However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's proof. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.
Marking the 70th anniversary of the original publication of Gödel's Proof, New York University Press is proud to publish this special anniversary edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.
Customer Reviews:
Godel's incompleteness theorems explained in non-technical language.......2007-09-20
There is no question in my mind that the most misunderstood mathematical theorems of all time are Godel's incompleteness theorems. In essence, they state that no system powerful enough to do basic arithmetic is complete. Meaning that there will always be statements that are true in the system but that can never be proven in that system. These results have been seized upon by people with many different agendas and used to argue conclusions as significant as the existence of God and that human intelligence is not simply the outward manifestation of neurotransmitters flowing from place to place. While this is all somewhat amusing, it is also disquieting, as the theorems cannot be used to conclusively justify such significant claims.
This book is one of the very first books where an attempt is made to explain Godel's theorems to the mathematical laity. In that sense, it is a success; the appropriate background is effectively put forward before the theorem and proof are explained. There is little in the way of formal mathematics and the bulk of the terminology is non-technical. If the people who use Godel's results to justify their extravagant claims were to read this book with an open mind, they would recognize the absurdity of their positions.
Godel's incompleteness theorem, clearly explained.......2007-06-28
Gödel's incompleteness theorem remains one of the most quoted, yet most misunderstood work in mathematics of the last century. Many non mathematicians had used the theorem (without understanding it, of course), to "prove" just about everything (generally it's been used to imply the limits of science, or stuff to that effect). The theorem, though, hardly implies that. This short book, written some 50 years ago, remains probably the best explanation of Godel's work available to the layman. The book starts explaining the background to Godel's theorem, as mathematicians such as Hilbert and Russell sought the axiomatization of mathematics. Godel's work, of course, proved that to be impossible. The book then proceeds to explain the theorem itself, as clearly as it possibly can (though I have to say that, as a non expert, the Gödel numbering scheme seems a like a trick to me, a sleight of hand. Yet, what do I know about this?). Overall, a great book about a much misunderstood work.
how i understand Godel.......2007-04-30
Godel was able to construct a formula from the axioms of Principia Mathematica (PM, and related systems, due to Russell and Whitehead) that is (roughly) "There exists no proof for this formula". This IS the formula itself. now, we need to know if this is true or not. so we try to find a proof for it within PM. if it is possible to find the proof, that means the formula is correct. but the formula says you can't find it. if the proof cannot be found, that means the negation of the formula is correct, but then the formula tells you that you cannot find it, so the formula must be correct, not its negation. in another words, the formula is undecidable within PM. also meaning PM is incomplete. what is amazing now is, of course that is what the formula tells you. so we do know it is true eventhough we can't show it within PM.
note that some people like reviewer Paul Vjecsner, who has also posted his disagreements on other Godel-related books, still confuse mathematics and meta-mathematics. although i have described and mixed-up meta-mathematics meanings to the formula above, Godel's proof was completely mathematicized within PM. either Vjecsner didn't understand the proof or he underestimated the grand aim of PM and thus the significance of Godel's work in taking it apart. Vjecsner argues that Godel's proof is essentially a linguistic paradox that has been unreasonably translated into PM. but the proof doesn't need that. that explanation is only done to give the readers a vague glimpse of the proof in a meta-mathemtical level. the proof only shows that there is a formula which is decidable if its negation is decidable. if you then argue that this is an unacceptable formula, then you are making even bigger claim than Godel, namely that the system is inconsistent! but the problem for you is you can't prove that within PM, thus still showing that PM is incomplete. Godel's proof is only as meaningful as PM. if you poke hole at Godel, you are poking hole at PM, which is exactly what Godel wanted to prove.
the book gives you an outline of how Godel went about constructing that formula with the language of PM, how he made the proof number-theoretical, and many more details. of course reading Godel's original paper would still be nightmarishly difficult even for many mathematicians, so the book gives a very good 'Godel's proof for dummies'. so you want to know what that formula looks like? read this book.
This is a strongly dissenting review.......2007-04-25
My challenge would no doubt be viewed with contempt, despite the many justified challenges of the status quo in history. Somehow it is always thought that although profound mistakes were made in the past, the contemporary establishment has things straightened out. My contention is that what is called Goedel's proof is a fundamental embarrassment in the history of logic and mathematics.
There are many particulars in this book that I can criticize which extend beyond Goedel. There is the formalization of deductive systems tracing back to Hilbert. It consists in using symbols without meaning, which are afterward "interpreted" according to subject matter. The like happened most prominently in non-Euclidean geometry, when terms like "straight line" and "plane" were reinterpreted to designate certain curved lines and surfaces, to then demonstrate that some Euclidean theorems do not hold. The problem is that this commits the basic fallacy of equivocation. By redefining words, the statements containing them no longer apply to the original meanings. Likewise, meaningless symbols in formal systems can be defined in conflicting ways. Presently this concerns the use of "Goedel numbers" for such symbols.
"Goedel's proof" is about a sentence which may be phrased as
"THIS STATEMENT IS UNPROVABLE (IN THE SYSTEM)".
The sentence has been recast into logical symbols, and it is these symbols, individually or combined, that are given "Goedel numbers". The argument then is that whatever is determined about the sentence applies to the numbers, viz. to mathematics. The equivocation should be obvious, but alas it isn't. The thinking is explained away by examples like the use of algebra in geometry. But these concern actual mathematical content. The sentence now at issue has no such content, the numbering merely designating the linguistic constituents.
Now to the sentence itself. As indicated there inside parentheses, the sentence is said unprovable in whatever deductive system is used, but Goedel is held to have proved it via "meta-mathematical" means outside the system. This contention I see as an excuse, avoiding contradiction. For, whatever the logic in the proof, it can be made part of the system. To exclude valid logical procedures from deductive systems unjustifiably cripples the systems, and to do so only serves the purpose of claiming an impossibility by not utilizing available tools.
This means that the alleged unprovability in logic, let alone mathematics, of statements otherwise decided true is false, and the statements can thus easily be shown to harbor contradictions and accordingly constitute paradoxes, rather than be part of an assumed consistent system. Regarding the above capitalized sentence, the argument advanced can be put as follows.
If the statement were provable then its negation,
"THE STATEMENT IS PROVABLE",
would also be provable, since that is what the provability would mean, and if this negation were provable then the original statement would also be provable, since the negation says so (p.99 in the book, in different words).
Then it is argued (p.100) that since "if a formula and its...negation can both be derived in some formal calculus [deductive system], then the calculus is not consistent", it follows that if it is consistent, "neither the formula...nor its negation can be demonstrable".
(This is said to make the formula "undecidable", which interestingly supposes the system consistent, in saying, p.101, "that an undecidable formula can be constructed within [it]". Here we see another pitfall of "formal", i.e. meaningless, systems. It is somehow thought that if they comply with grammatical rules, there is no problem. But grammatical sentences can in meaning be contradictory.)
It is further argued (pp.101-2) that one of the above capitalized statements must be true, and since the initial one was found "undecidable", namely to have "no proof inside [the system]", it is the true one, by asserting that unprovability.
It is finally reemphasized that this was not deduced "from the axioms and rules of a formal system, but by a meta-mathematical argument."
But those axioms and rules, unlike contended, easily accommodate the logic, which is quite common. That neither of the above capitalized sentences is provable because it implies the other follows by the "law of contradiction"; that one of them must presumably be true follows by the "law of excluded middle"; and that it is the initial sentence which is true, since just found unprovable, can be said to follow by the "law of identity". Thus the unprovable sentence is contradictorily proved via the well-known "laws of thought".
As indicated, the sentence is correspondingly internally contradictory and can be added to other paradoxes, rather than be proclaimed true and unprovable.
Interesting introduction and some very good hints to a conclusion.......2007-04-18
I have been a big fan of the issue of formal mathematics and the theory of computation but I always missed a full grasping of the goedel theorem. The book presents the line along which Goedel moved: mostly formal systems and the most interesting issue of "calculating grammar" by powers of primes: an outstanding example, to me, was how using that instrument one could know whether a sentence was introduced by a "not" or not (in the case simply by checking if the figure expressing the formula was even or odd).
In that way the full system of mathematics turned into a sort of a computer program that of course could not calculate every function.
For that matter I came to wonder why the demonstration of goedel theorem could not be carried out simply by showing a formal system has the same power as a universal Turing machine and thus transferring to it the (much easier) results obtained on that issue - like for example to problem of the stop or the one of finding any semantic information in a program without actually executing it.
Amazon.com
Kurt Gödel is often held up as an intellectual revolutionary whose incompleteness theorem helped tear down the notion that there was anything certain about the universe. Philosophy professor, novelist, and MacArthur Fellow Rebecca Goldstein reinterprets the evidence and restores to Gödel's famous idea the meaning he claimed he intended: that there is a mathematical truth--an objective certainty--underlying everything and existing independently of human thought. Gödel, Goldstein maintains, was an intellectual heir to Plato whose sense of alienation from the positivists and postmodernists of the 1940s was only ameliorated by his friendship with another intellectual giant, Albert Einstein. As Goldstein writes, "That his work, like Einstein's, has been interpreted as not only consistent with the revolt against objectivity but also as among its most compelling driving forces is ... more than a little ironic."
This and other paradoxes of Gödel's life are woven throughout Incompleteness, with biographical details taking something of a back seat to the philosophical and mathematical underpinnings of his theories. As an introduction to one of the three most profound scientific insights of the 20th century (the other two being Einstein's relativity and Heisenberg's uncertainty principle), Incompleteness is accessible, yet intellectually rigorous. Goldstein succeeds admirably in retiring inaccurate interpretations of Gödel's ideas. --Therese Littleton
Book Description
A masterly introduction to the life and thought of the man who transformed our conception of math forever.
Kurt Gödel is considered the greatest logician since Aristotle. His monumental theorem of incompleteness demonstrated that in every formal system of arithmetic there are true statements that nevertheless cannot be proved. The result was an upheaval that spread far beyond mathematics, challenging conceptions of the nature of the mind.
Rebecca Goldstein, a MacArthur-winning novelist and philosopher, explains the philosophical vision that inspired Gödel's mathematics, and reveals the ironic twist that led to radical misinterpretations of his theorems by the trendier intellectual fashions of the day, from positivism to postmodernism. Ironically, both he and his close friend Einstein felt themselves intellectual exiles, even as their work was cited as among the most important in twentieth-century thought. For Gödel , the sense of isolation would have tragic consequences.
This lucid and accessible study makes Gödel's theorem and its mindbending implications comprehensible to the general reader, while bringing this eccentric, tortured genius and his world to life.
About the series:Great Discoveries brings together renowned writers from diverse backgrounds to tell the stories of crucial scientific breakthroughs—the great discoveries that have gone on to transform our view of the world.
Customer Reviews:
Extraordinarily bad.......2007-09-22
A man is reading a book. For an unknown reason, the book begins in the most execrable of tenses, the historical present. He ponders putting the book down, but continues reading in the hope that both the literary style will improve, and that the ostensible topic of the book - Godel's work on incompleteness - will be taken up, in preference to biographical information about Einstein. Some 40 pages later, his hopes are still not met. He begins reading the next chapter, his hopes buoyed by the book's title, which is, after all, Incompleteness. He frowns upon discovering that this chapter is still not about Godel's work, but about the boy. He reads sentences like "Kurt Godel fell in love with Platonism" and "I found... those little Bible studies published by the Jehovah's witnesses, the kind that their itinerants will urge on you if you happen to be home in the middle of the day and answer the door." He thinks "I don't give a fig about Rebecca Goldstein's discoveries of scraps of paper once drawn on by Godel. Get to the point." The book does not get to the point. He reads "First exposure to Plato can be an extremely heady experience for those with a passion for abstraction. (I remember my own.)" He thinks "this woman has *no* passion for abstraction, she's trying to psychologise mathematical logic!" He reads pages and pages about Vienna and Karl Kraus, an asinine attempt to discuss the logical positivists as *subjectivists* of all things. He reads about the villainous Wittgenstein. He thinks, "what a terrible interpretation of Wittgenstein." By now the book has passed into the past tense, thank God. I finally realised what was going on: Rebecca Goldstein was writing this book in the hope that it would get picked up and adapted as a screen-play. Thus, she needed a love story (Godel meets Platonism; they have wonderfully incomplete children and live neurotically ever after); she needed a villain ('subjectivism,' which, as far as i can tell, means everything which isn't Platonism in this tale, and wears a black hat while shooting up bars and dishonoring ladies); she needed to avoid any sort of philosophical, mathematical or historical rigor at all costs.
This is a long review, and probably boring. It is a performative review. Buy Nagel's book, 'Godel's Proof,' instead.
A disappointment due to its incompleteness .......2007-06-03
This is a good companion and counterpoint to "Wittgenstein's Poker" that starts off well but ends as a book with no payoff. Other than her well-argued characterization of Godel as a Platonist among positivists whose ideas were misunderstood or ignored, Goldstein presents neither coherent biography, nor any explanation of the development and significant influence of Godel's work after 1931 (a subject that seems beyond Goldstein's capabilities). Despite an occasional mention of an important date and a few details of political intrigue at the Institute for Advanced Study in between, the book has almost no content about Godel between 1931 to the 1970's, not long prior to Godel's starving himself to death.
Goldstein does present a decent overview of the first two incompleteness theorems and the goals of the formalists who preceded Godel, though the view she presents is very limited because it ignores the issues of pervasive errors in mathematical reasoning about the infinite in analysis and other fields that led to the formalist point of view in the first place. She is somewhat fuzzy, though, on the relationsip between completeness and incompleteness. There are some obvious errors in terms of the surrounding explanations from small details - Hilbert presented 23 major problems, not 10, at the 1900 Math Congress - to a misleading implication that the theory of arithmetic underlies all of mathematics and its incompleteness implies that the formal inference no value for complex mathematical domains (ignoring efficiently decidable theories like that of real-closed fields, for example).
As other reviewers have noted, Goldstein has almost nothing to say about Godel's relationship with von Neumann who has his biggest champion. Despite her mention of Turing's work, other than saying that Godel was pleased by it she seems to describe it as little more than changing the terminology of Godel's work. Finally, she repeats the Lucas argument and Godel's lack of sanction for it, but she does not seriously discuss the refutations of the Lucas argument that have appeared. The focus of her text seems to imply that, in his heart of hearts, Godel would have liked to believe it.
Both technically and philosophically confused........2007-04-21
As a student of logic and the philosophy of mathematics I found this book seriously confusing. The book contains a number of technical, biographical and philosophical errors. Interested students should try and get hold of reviews of "Incompleteness" by Sol Feferman and Juilette Kennedy, logicians who explain the book's flaws better than I could.
Is a Theory of Everything really possible?.......2007-03-10
Absolutely fascinating. Not only that there are things we cannot know, but we can even know there are things we cannot know.
Read this review: don't buy this book!.......2007-01-17
I bought this book in the naive supposition that it explains incompleteness theory. It does not. It is a mix of some Gödel biography and a lot of philosophy. Philosophy not about Gödel only. If you like philosopy, OK. But from this book you will learn no formula, no math, no explanation of any scientific value of incompleteness theory. This book is a dud. Don't buy it.
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Types, Tableaus, and Gödel's God (Trends in Logic)
M. Fitting
Manufacturer: Springer
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Incompleteness in the Land of Sets (Studies in Logic)
ASIN: 1402006047 |
Book Description
Gödel's modal ontological argument is the centrepiece of an extensive examination of intensional logic. First, classical type theory is presented semantically, tableau rules for it are introduced, and the Prawitz/Takahashi completeness proof is given. Then modal machinery is added, semantically and through tableau rules, to produce a modified version of Montague/Gallin intensional logic. Extensionality, rigidity, equality, identity, and definite descriptions are investigated. Finally, various ontological proofs for the existence of God are discussed informally, and the Gödel argument is fully formalized. Objections to the Gödel argument are examined, including one due to Howard Sobel showing Gödel's assumptions are so strong that the modal logic collapses. It is shown that this argument depends critically on whether properties are understood intensionally or extensionally.
Parts of the book are mathematical, parts philosophical. A reader interested in (modal) type theory can safely skip ontological issues, just as one interested in Gödel's argument can omit the more mathematical portions, such as the completeness proof for tableaus. There should be something for everybody (and perhaps everything for somebody).
Book Description
This breakthrough book establishes deep connections between elementary particle theories such as the Standard Model and Superstring theories, and computer languages such as Assembly language, C, and C++ suitably extended. It also proves, for the first time, that the universe must be quantum in nature based on Godel¿s celebrated theorem (that there are statements in any non-trivial mathematical deductive system that cannot be proved or disproved.) Therefore all attempts at building a deterministic fundamental theory of physics (such as Bohm¿s theory) are unacceptable.
Customer Reviews:
Excellent and professional service.......2005-08-13
The book arrived promptly, and it was in excellent shape. I was very pleased with the professional service I received.
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Computational Logic and Proof Theory: 5th Kurt Gödel Colloquium, KGC'97, Vienna, Austria, August 25-29, 1997, Proceedings (Lecture Notes in Computer Science)
Manufacturer: Springer
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Binding: Paperback
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ASIN: 3540633855 |
Book Description
This book constitutes the refereed proceedings of the 5th Kurt Gödel Colloquium on Computational Logic and Proof Theory, KGC '97, held in Vienna, Austria, in August 1997.The volume presents 20 revised full papers selected from 38 submitted papers. Also included are seven invited contributions by leading experts in the area. The book documents interdisciplinary work done in the area of computer science and mathematical logics by combining research on provability, analysis of proofs, proof search, and complexity.
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Godels Proof
Ernest Nagel
Manufacturer: NY UNIV+PRESS
ProductGroup: Book
Binding: Hardcover
ASIN: B000QA7WNC |
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A fascinating account of well-chosen episodes in Godel's life together with an accessible account of Godel's extraordinary achievement - basically a proof that there are true but unprovable statements.
Book Description
This volume is a translation of the book Gödel, written in Japanese by Gaisi Takeuti, a distinguished proof theorist. The core of the book comprises a memoir of K Gödel, Takeuti's personal recollections, and his interpretation of Gödel's attitudes towards mathematical logic. It also contains Takeuti's recollection of association with some other famous logicians. Everything in the book is original, as the author adheres to his own experiences and interpretations. There is also an article on Hilbert's second problem as well as on the author's fundamental conjecture about second order logic.
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