Product Description
Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable."His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame.In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems.The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic.As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists. ... Read more

Customer Reviews (5)

A clear presentation of Godel's proof
Unlike most other popular books on Godel's incompleteness theorem, Smulyan's book gives an understandable and fairly complete account of Godel's proof. No longer must the undergrad fanboy be satisfied in the knowledge that Godel used some system of encoding "Godel numbers" to represent a metamathematical statement with a mathematical one. The power of the proof can now be yours!

Seriously though, among the large family of well written accounts of Godel's theorem, including Godel, Escher, Bach, as well as Nagel and Newman's book, Smulyan's is the most direct and serious account, and accessible to anyone with the mathematical maturity to handle an advanced level undergraduate math class.

Well Worth The Investment

J Alfonso, in a review of Dover's On Formally Undecidable Propositions of Principia Mathematica and Related Systems, wrote:

You do not wade through difficult theorems in mathematical logic without the appropriate tools. And the appropriate tools include having done similar but simpler proofs on your own and having a solid background in mathematical logic. Without this background, it doesn't matter whether you have the ability to be a mathematics professor at Princeton or place top five in the Putnam - you simply will not understand the proof in a rigorous manner.


It was in that review that he suggested interested readers get a copy of Smullyan's book. So I did. I was hesitant because it was expensive, but in the end if you really do want a complete and thorough understanding of Gödel's proof then this is maybe the most efficient way to go about it.

It is a challenging but accessible book that requires little prerequisite knowledge. A basic familiarity in first order logic is all that is required. And quite frankly if you can't make it through this book then I don't understand how you expect to make it through Gödel's actual proof.

I believe that both Smullyan's book and Gödel's proof can be understood by any diligent student, so don't take my last comment to suggest this it's an impossibly difficult task. Rather what I am trying to suggest is that if you really want to understand Gödel's proof then this book may or may not be easy for you, but it certainly will be tremendously valuable and is worth the effort.

A very good book but requires correction of typos
Raymond Smullyan is a logician that I admire much.This book is very good but contains many typos and mistakes.For example, in p.31, the definition of xPy does not work as it is not able to account for 0P305.The definition should be corrected as:
xPy iff There is z not greater than y (zBy and xEz).Similar mistakes can be found elsewhere.And this book thus requires another edition for the coorection of typos and mistakes.

Finally -- Straight Talk About Incompleteness!
Well. This is the book. Read this instead of, or before you read Goedel�s paper. Within 20 pages you will know the �trick� that Goedel used. It�s a beauty, but it is far easier to see it under Smullyan�s tutelage than by coming to the classic paper cold, since Goedel uses a more difficult scheme to achieve his ends. Much work has been done since 1931, and we get the benefit of the stripping-down to essentials that such as Tarski (and Smullyan himself) have contributed.

The book has much of interest to those who wish to pursue the subject of the incompleteness and/or consistency of mathematics, or to come at Goedel from a number of angles. For me, though, the first 3 chapters were enough. I just wanted to find out how K.G. did what he did. Now I know, and I know where to go if I need even more.

The exercises are helpful to keep you on track and test your understanding. They also contribute materially to the exposition. A stumbling-block for many readers will be the extremely abstract nature of the discussion, and the new notations and definitions that constantly come at one. Viewing numbers as strings and strings as numbers (and knowing when to switch from one view to another) will be confusing at first. This is the hard part: what Goedel did, in essence, is demonstrate that one can view proofs in two ways � as numbers, and as strings of characters. As in viewing an optical illusion, it is sometimes tough to hold the proper picture in mind.

Smullyan�s book �First-Order Logic� is enough preparation for this work. One must here, even more than there, keep straight the difference between the �proofs� that are part of the subject matter (and so are strings of characters), and the proofs we go through that verify facts about these strings. Before we started reading this book, of course, we had some informal sense that we were going to prove something about proofs. What we are REALLY doing, though, is proving something about �proofs�. You get the picture. Goedel must have been a lot of fun at parties.

Mainline Incompleteness with this Book!
I highly recommend this title because it supplys all the necessary proofs for a nuts and bolts understanding of incompleteness, including incompleteness proofs for Peano arithmetic and the unprovability ofconsistency.

This title is a difficult read but the only prerequisite isa familiarity of first-order logic equivalent to a one semester collegecourse.

A lot of the proofs are based on new material and are easier tounderstand than the original work by KG.

An added benefit is theexercises.They are not impossible and aid in one's understanding.

This book is well worth the work in demands. ... Read more

Product Description
Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments.

• Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters
• Discusses interpretations of the Theorem made by celebrated contemporary thinkers
• Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories
• Written in an accessible, non-technical style

Customer Reviews (2)

Enhanced my understanding
I've often wondered how professional film critics can figure out how much of their reaction to a movie is due to its intrinsic merits and how much is based on the mood they happened to be in when they saw it. I find myself in an analogous situation as I write this review.I read _Godel, Escher, and Bach_ as a college freshman when it first came out, I've read Franzen's book on Godel, and I've been exposed to Godel's arguments on several other occasions during my career, but reading _There's Something about Godel_ has left me with a much better understanding of the Incompleteness Theorem than I previously had.I don't know for sure how much of this is really due to Berto's skill as a thinker and a writer and how much is due to those past influences finally sinking in, but to counterbalance some of the criticism he's received from others, I'm willing to give him the credit.He impressed me as a very clear writer, being patient without being tedious.(In that last aspect, this book didn't feel at all like a pop math book, a genre which professional mathematicians typically find boring.)Even the chapter on Wittgenstein, which I anticipated hating, was surprisingly tolerable, though I don't think I'm in any danger of becoming a fan of either Wittgenstein or paraconsistent logic.

Two final points:

(1)This is an English translation of an Italian book, and I presume Berto is a native Italian, but the English in this book is just fine--not at all stilted.
(2)Wiley has come out with some books with really ugly printing lately, even uglier than an average print-on-demand book.There's no such problem with this book, though.

A perpetuation of a fundamental falsehood
This is not my first review of a book on Gödel, and therefore I want to respond freshly, by considering this book in particular.

In view of the numerous attacks I was subjected to for not accepting "Gödel's incompleteness theorem", the main subject of this book, I could be taking comfort in author Berto's attention to Wittgenstein as a critic of Gödel's theorem. At least I have that prominent figure sharing my negative attitude, with Dr. Berto giving Wittgenstein some credence (p.213): "an audacious rethinking...may nowadays vindicate some of Wittgenstein's 'outrageous claims' on Gödel's Theorem, too swiftly dismissed by commentators who dogmatically took the logic of Russell and Frege as the One True Logic".

My view of Wittgenstein, though, is not the best, since he seems never to substantiate his flamboyant declarations, and Dr. Berto is really fully committed to Gödel, describing him, in keeping with today's adulation, as "the logician of the [last] millennium" (p.189).

In my eyes Gödel is nothing of the sort. He contrived an impossibly elaborately symbolized "proof" of a simple sentence, and perhaps still more preposterously he equivocated that sentence with a mathematical one. That sentence, famous by now, is

(1) "THIS SENTENCE IS UNPROVABLE" (in the logical system in which it occurs).

The sentence is modeled on the ancient Liar paradox, stating

(2) "THIS SENTENCE IS FALSE".

If (2) is true then, by its content, it is false; and if it is false then, again by its content, it is true.

Now, Dr. Berto cites logician Alfred Tarski as showing that the truth of a sentence cannot be defined in the language of the sentence (similarly to the above unprovability), quoting his statement (p155): "'Snow is white' is true...if and only if snow is white". The single quotes delimit such a sentence, the rest quoted is to belong to the "metalanguage". But why the distinction? Because the Liar, sentence (2), couldn't be resolved for over two millennia. However, its resolution is very simple. By the truth of a sentence, expressed in the same language, we mean what it says. The Liar, like any sentence, is meant to be true, but illegitimately says it is false; i.e. it harbors an implicit contradiction, same as if a sentence explicitly declares something both true and false. (Allow me to mention I treat this and other issues in my book "On proof...")

But as a result of the Liar and other paradoxes, all internally contradictory, logicians instead devised various hierarchies, of languages, types, systems, by which the contradictory results would be eliminated by placing them in other hierarchies. One such attempt was made by David Hilbert (p.39ff), who famously proposed "formal systems" of logical symbols lacking meaning and "interpreted" afterward, delegating the logical process to a separate system of "metamathematics". This is fundamentally the basis of Gödel's arguments.

He made his above sentence (1) an "interpretation" of (giving meaning to) certain symbols that are part of a formal system, thereafter arguing that his proof is achieved via separate metamathematics. But is there such a distinction between a logical system and a proof regarding it? Logic in general consists of principles to be followed, such as the "laws of thought", referred to in the book (p.7) by the "Principle of Bivalence" ("all sentences are either true or false") and the "Law of Non-Contradiction" (a sentence cannot be both true and false). Accordingly, if such logic is applied to sentences, it is applied within the system. Now consider sentence (1), the one alleged proved.

As Dr. Berto offers the proof in one case (p.50), "Suppose [(1)] is provable. Then, given what it says, it is false, since it claims not to be provable... Therefore [(1)] is not provable in [the system]... But if [(1)] is not provable, then [(1)] is what it claims to be; therefore, it's a true sentence". But wait! How did we reach this conclusion? By finding the provability of [(1)] a contradiction, thereby PROVING within the system the sentence true! But by thus proving it, we again contradict it! The upshot is, sentence [(1)] ends up a paradox.

As indicated, the finding of paradoxes in logical or mathematical systems is thought intolerable, and therefore great efforts have been expended to make the systems paradox-free, largely by devising mentioned hierarchies. But the paradoxes are no reflection on the systems; they are contained within the particular sentences or descriptions concerned. Exceptions can be paradoxes concerning whole theories, like Cantorian set theory. I can't go into more detail here, adding only a remark again on allegations that Gödel's theorem applies to just mentioned mathematics.

"Gödel numbering" is an impressively intricate system of assigning numbers to constituents of "formal systems", like the symbols for sentence (1). It is then alleged that (1) also applies to some of mathematics. But however imposing the numerical names of the linguistic constituents, of the symbols, they don't change the contents behind them. I.e., to change the content of sentence (1) into a numerical one commits the fallacy of equivocation.
... Read more

Product Description
The question "What is truth?" is a question that Ralph McNeil, a down-to-earth engineering student, has never before had cause to ponder. But when he encounters a famous theorem (Gödel's 'Incompleteness Theorem') which poses a seemingly unsolvable puzzle about the very meaning of truth itself, he becomes convinced against all the prevailing wisdom that the theorem must be wrong, despite the fact that it has been accepted as correct for over seventy-five years. He sets out on a journey to prove it wrong, a journey that becomes a quest to seek out the truth about truth itself. As he pursues his mission, we see into the heart and mind of Kurt Gödel, the enigmatic character who first posed the theorem - a man who won acclaim as the greatest logician of his time, but a man tortured by his own deep mystical convictions, convictions that would eventually drive him to the brink of insanity. Fact and fiction, past and present are intertwined in a compelling story of love, intrigue, deception and death that leads to a startling conclusion that will change forever the way the world understands the concept of truth. ... Read more

Customer Reviews (1)

A COMPLETE MISUNDERSTANDING OF GÖDEL'S THEOREM
I must confess this was a pleasant read although I became annoyed by the absurd claims the author and publisher makes about Gödel's theorem by means of the novel's main character, the student Ralph McNeil.

Contrary to what he claims in the preface, the author is essentially misguided about Gödel's famous result and his contention to have disproved it is ridiculous. There are errors even in his exposition of the theorem.

So, the reader may really enjoy the novel but should not take the mathematical claims of its main character too seriously. My advise: take it as complete fiction and enjoy it as such.

... Read more

Product Description
This thoroughly revised second edition of a classic book on the main ideas and results of general meta-mathematics contains new results and simplified proofs, as well as an up to date bibliography. In addition to the standard results of Gödel and others on incompleteness, (non) finite axiomatizability, interpretability, etc.., it contains a thorough treatment of partial conservativity and degrees of interpretability. The reader should be familiar with the widely used method of arithmetization and with the elements of recursion theory. ... Read more

Customer Reviews (1)

Solid Introduction to Logic
This book is well written and a good resource but nothing earth shattering. You'll learn a lot more my taking a good course in advanced logic. ... Read more

Product Description
Lecture Notes in Logic 10 is concerned almost exclusively with properties that are common to all sufficiently strong, axiomatizable theories. Paper. DLC: Incompleteness theorems. ... Read more

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Chapters: Gödel's Incompleteness Theorems. Source: Wikipedia. Pages: 342. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem shows that if such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself. In mathematical logic, a theory is a set of sentences expressed in a formal language. Some statements in a theory are included without proof (these are the axioms of the theory), and others (the theorems) are included because they are implied by the axioms. Because statements of a formal theory are written in symbolic form, it is possible to mechanically verify that a formal proof from a finite set of axioms is valid. This task, known as automatic proof verification, is closely related to automated theorem proving; the difference is that instead of constructing a new proof, the pr...More: http://booksllc.net/?id=58863 ... Read more

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This introduction to mathematical logic takes Gödel's incompleteness theorem as a starting point. It goes beyond a standard text book and should interest everyone from mathematicians to philosophers and general readers who wish to understand the foundations and limitations of modern mathematics. ... Read more

Customer Reviews (4)

Corrected second printing exists
In the words of a faculty member "this first printing is so riddled with typos and grave mathematical errors that you would have to include a multi-page errata sheet with it to make it usable."There is no indication in any listing for the book that a corrected printing exists.We called the publisher to insure we were getting the second printing, but at least one patron lamented that they had purchased a copy online and ended up with the first printing.The only indication of changes in the 2nd printing is a single sentence on page xiii thanking the faculty member for submitting corrections over the first printing. PAMNET@listserv.nd.edu 9/20/07

Let it go out of print
Although it was meant for a one-year course, this text lacks much basic material. There is no set theory, no second-order logic, and almost nothing on recursion. Many basic concepts are given cursory mention. In fact, nothing seems to be developed as fully as one normally sees in an introductory text. The whole book seems like a jumble of information rather than a coherent narrative. I pity the students who had to use it.

Nope, didn't like it.
This book is an introduction to mathematical logic, covering the syntax and semantics of propositional and first-order logic, the Hilbert-style proof system and its completeness, some model-theoretic material, and Godel's (first) incompleteness theorem. Its more formal and rigorous than most introductory books, which is the style I prefer, but I was left feeling unsatisfied with the book. It was hard to nail down exactly what I didn't like about it, but what I came up with is this: although the theorems and proofs are ok, considered one at a time, the overall perspective of what's going on and how things relate to each other was left hazy. Perhaps better exposition and historical background would correct this, but I found the book unsuitable for self-study for a beginner. This was where I first learned Godel's incompleteness theorem, and even though the version presented is particularly weak (Peano arithmetic is incomplete), I was left confused about the significance of the theorem and exactly what assumptions were used in the course of the proof. I see now that their attempt at simplification is what led to my misunderstandings. If you're looking for a good general mathematical logic book, I seriously recommend you get Enderton instead (see my reviews). If you want a book focused on the incompleteness theorems get Smullyan's excellent GIT.

A good simplification for teaching
This is an interesting simplification of Goedel's first incompleteness theorem. The book assumes there is a standard model of the Peano axioms, so that in effect it assumes the axioms consistent and even true. Then it shows nonetheless the axioms cannot decide every sentence. This brings out the main point of incompleteness, I think. But it is far weaker than Goedel's proof in two ways: it uses stronger assumptions, and it proves incompleteness only for the standard Peano axioms. There is a brief discussion of how this kind of proof would work for any consistent extension of the Peano axioms but I did not find it very helpful to my class. Of course the assumptions pay off in a very much quicker proof and much less concern with syntax.

In fact, the main problem with the book is that the assumptions are never made quite clear. The authors say several times that their result is weaker than Goedel's, but never say why. A discussion of this somewhere in the book would be helpful--both to students and profs. Presumably they do not use the full strength of a standard model of the Peano axioms, but it would be a chore to go through and see just what they do use.

I taught this book as a one semester course for students who had previously seen predicate logic in an intro course. To fit it into a semester I skipped the chapter on model theory (not needed for the incompleteness theorem) and the material at the end on recursive functions. The book gives a very pretty account of induction, stressing from the start that the natural numbers are just one case of an inductive structure. This made later inductions on, say, well-formed formulas, very clear to the class. The students got the compactness theorem very easily, as they had not in other class I've taught from other books. The short account of non-standard models for arithmetic is helpful in showing that Goedel's theorem is *not* about whether the Peano axioms say all there is to know about arithmetic--the fairly simple compactness theorem already shows no first order theory can do that. I expect to use this book again the next time I teach the subject. ... Read more

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High Quality Content by WIKIPEDIA articles! Typographical Number Theory (TNT) is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach. It is an implementation of Peano arithmetic that Hofstadter uses to help explain Gödel's incompleteness theorems.Like any system implementing the Peano axioms, TNT is capable of referring to itself (it is self-referential).The symbol S can be interpreted as "the successor of", or "the number after". Since this is, however, a number theory, such interpretations are useful, but not strict. We cannot say that because four is the successor of three that four is SSSS0, but rather that since three is the successor of two, which is the successor of one, which is the successor of zero, which we have described as 0, four can be "proved" to be SSSS0. TNT is designed such that everything must be proven before it can be said to be true. This is its true power, and to undermine it would be to undermine its very usefulness. ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Entscheidungsproblem, Gödel's Completeness Theorem, Compactness Theorem, Gödel's Incompleteness Theorems, Löwenheim-skolem Theorem, Deduction Theorem, Tarski's Undefinability Theorem, Herbrand's Theorem, Gentzen's Consistency Proof, Löb's Theorem, Metatheorem, Lindström's Theorem, Barwise Compactness Theorem. Excerpt:In mathematical logic , the Barwise compactness theorem , named after Jon Barwise , is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967. Statement of the theorem References (URLs online) Websites (URLs online) A hyperlinked version of this chapter is at In mathematical logic , the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory , as it provides a useful method for constructing models of any set of sentences that is finitely consistent . The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces ; hence, the theorem 's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along with the downward Löwenheim Skolem theorem , that is used in Lindström's theorem to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them. Applicat... ... Read more

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In algorithmic information theory, the Kolmogorov complexity of an object such as a piece of text is a measure of the computational resources needed to specify the object. The first string admits a short English language description, namely "ab 32 times", which consists of 11 characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, which has 64 characters. More formally, the complexity of a string is the length of the string's shortest description in some fixed universal description language. The sensitivity of complexity relative to the choice of description language is discussed below. It can be shown that the Kolmogorov complexity of any string cannot be too much larger than the length of the string itself. Strings whose Kolmogorov complexity is small relative to the string's size are not considered to be complex. The notion of Kolmogorov complexity is surprisingly deep and can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem. ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 926 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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This digital document is an article from Encyclopedia of Philosophy, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 9423 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Explores major marketing and advertising campaigns from 1999-2006. Entries profile recent print, radio, television, billboard and Internet campaigns. Each essay discusses the historical context of the campaign, the target market, the competition, marketing strategy, and the outcome. ... Read more

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This digital document is an article from Encyclopedia of Science and Religion, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 337 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Addresses the interactions, contradictions, and tensions between science and religion, both historically and in contemporary life. The set examines technologies like in vitro fertilization, cloning, and continuing developments in neurophysiology against the backdrop of deeply-held religious beliefs. In addition, phenomena such as the Church of Scientology are also studied, along with more traditional issues, such as the origins of life, the nature of sin, and the philosophy of science and religion. ... Read more