Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse 1st Edition by Torkel Franzén (PDF)

Description

Among the many expositions of Gödel’s incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature. — John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt Gödel

User’s Reviews

Editorial Reviews: Review ” “”Franzén’s book is accessible, well written, and often funny…”” -Richard Zach, History and Philosophy of Logic, July 2005 “”Ich möchte allen meinen Lesern . . . ein Buch ans Herz legen, und zwar “”das Neue”” von Torkel Franzén: Gödel’s Theorem – An Incomplete Guide to Its Use and Abuse…”” -Altpapier, October 2005 “”If the reader is serious about understanding the scope and limitations of Gödel’s theorems, this book will serve them well.”” -Don Vestal, MAA Online, November 2005 “”. . . This is an excellent book, carefully considered and well-written. It will be read by layman and expert alike with pleasure and profit.”” -Peter A. Fillmore, CMS Notes, Volume 37 No. 8, December 2005 “”… a welcome tourist’s guide not only to the correct but also to many incorrect interpretations of the theorems, both in their immediate contexts and in wider circumstances.”” -I. Grattan-Guinness, LMS, February 2007 “”This is a marvelous book. It is both highly competent and yet enjoyably readable. … At last there is available a book that one can wholeheartedly recommend for anyone interested in Gödel’s incompleteness theorem―one of the most exciting and wide-ranging achievements of scientific thought ever.”” -Panu Raatikainen, Notices of the AMS, February 2007 “”This is a marvelous book. It is both highly competent and yet enjoyably readable. … At last there is available a book that one can wholeheartedly recommend for anyone interested in Gödel’s incompleteness theorem―one of the most exciting and wide-ranging achievements of scientific thought ever.”” -Panu Raatikainen, Notices of the AMS, March 2007 “”… an extraordinary addition to the literature. … The book is ideal reading for people with a basic logical background, be they computer scientists, philosophers, mathematicians, physicists, cognitive psychologists, or engineers … and a real desire to understand quite deeply one of the intellectual gems of the 20th century.”” -Wilfried Sieg, Mathematiacl Reviews, March 2007 “”… lively and a pleasure to read … provides remarkably sharp formulations of the usual confusions. There is no doubt that readers of this journal should recommend this book to any friends or colleagues who ask about the ramifications of incompleteness.”” -Stewart Shapiro, Philosophia Mathematica, June 2006 “”Dawson’s biography of GÄodel is provocative and interesting on several fronts, and is highly recommended to anyone with an interest in logic, the foundations of mathematics or the history of mathematics.”” -Samuel R. Buss Buss, December 1998 “”This book presents an exceptional exposition of Gödel’s incompleteness theorems for non-specialists … a valuable addition to the literature.”” -EMS, March 2006 “”The book explains fully, without using any technical logical apparatus, Gödel’s two theorems about the incompleteness of any formal system which includes elementary arithmetic … It is a great success in the way that the proofs of the theorems, while not given in full, are outlined in sufficient detail to make a discussion of the different versions that have been given worthwhile. I do not think there is any non-specialist exposition comparable for clarity and thoroughness.”” -Clive Kilmister, The Mathematical Gazette, March 2007 “”Franzen touches upon contemporary issues in logic that otherwise only rarely find their way into books of an introductory character like this one.”” -The Review of Modern Logic, March 2007 “”Torkel Franzen’s “”Goedel’s Theorem”” is a wonderful book, destined to become a classic … In “”Goedel’s Theorem,”” Torkel Franzen does a superb job of explaining clearly and carefully what the incompleteness theorem says and its implications as well as skewering much of the nonsense that has been written about it. … However, while “”Goedel’s Theorem”” should be accessible to a general audience, “”Inexhaustibility”” may be rather rough going for a reader who has not seriously studied mathematical logic.”” -Mathematics and Comupter Science, March 2008″ About the Author A philosopher by training (PhD, University of Stockholm), Torkel Franzén has for the past twenty years been active working in computer science (at the Swedish Institute of Computer Science) and teaching programming (at Luleå University of Technology). He is the author of a number of books, among them Inexhaustibility: A Non-Exhaustive Treatment.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I really hate to give a book a bad review, however this book does not explain Godel’s theorem in a logical cohesive manner. This can be seen early on when the author defines an algorithm as “a purely mechanical, computational procedure, one that when applied to a given number or finite sequence of numbers always terminates” (12). Of course it always terminates, because the procedure is, by his definition, applied either to a given number or to a finite sequence of numbers, which makes it a finite procedure. He further defines a computable property as a “property of numbers that can be checked by applying an algorithm”(12). He then further defines a Goldbach-like mathematical statement as a statement that has a computable property (12). Please don’t lose me here, The author is clearly saying that algorithms are finite procedures, and because computable properties are (by his definition) properties that can be checked with algorithms they must also be finite. And because Goldbach-like statements are (by his definition) statements that have computable properties they must also be finite. Follow his explanation so far? If algorithms are finite, then computable properties are finite because they are based on algorithms. If computable properties are finite then Goldbach-like statements are finite because they are based on computable properties. Now get this, the author then makes the following non-sensical point regarding Goldbach-like statements:”If A is not provable in S, a systematic search will in general just go on forever without yielding any result. So for any formal system S that incorporates a bit of arithmetic – the basic rules needed to carry out computations – a Goldbach-like statement is disprovable in S if false. On the other hand, we cannot make any similar observations about how a Goldbach-like statement can be proved if it is true. For every n [meaning number], a computation can indeed verify that every number 0,1,…,n has property P, but this is not a verification that every number has property P….” (14)Please don’t lose me here. The author just made a condition for proving a Goldbach statement as true if and only if it is true for “every number”. However the set of every number is infinite. He has just contradicted himself. At first he defines Goldbach statements as statements that deal with finite sets of numbers and then claims they are not provable because they don’t apply to infinite sets. Of course they don’t apply to infinite sets, that’s how he defined them to begin with. The author needs to make up his mind. He must either define Goldbach statements as finite or infinite. This schizophrenia continues when he discusses non Goldbach statements.The whole book is written this way, and if your scratching your head now, imagine what reading the whole book would be like. It’s a disaster.Today is December 23, 2016. I am editing my original post (written around 5 years ago now) to reflect that I stand by my original post in that this book is not for you if you do not have a higher math and computer science background. Although I like what the author says later in the book regarding the abuse of Gödel’s theorem, (i.e. 4,5 and 6) my review is regarding how he explains the theorem, not what he says about the theorem. If you are looking to learn what Gödel’s theorem is all about and you are not a math or computer science buff, you will not like this book.

⭐Aside from my finding Goedel’s theorem false, I see this book as dismally failing in its purport to be written for a general audience, also contended in the two blurbs on the back cover, stating that the book “explain[s] clearly and thoroughly just what the theorems really say” and “With exceptional clarity…gives careful, non-technical explanations…”The book instead indulges in such a profusion of technical language that it appears only suitable for discussions in specialized journals, and indeed there seems to be a polemic going on in it with many fellow-professionals, including well known scientists like Hawking, Dyson or Penrose. In the process the author doesn’t as much as give a clear form of the basic “Goedel sentence”, around which the theorem revolves, although he refers to it numerous times, notably on page 55. Likewise he doesn’t make clear how “Goedel numbering” is performed, claiming it a “technicality that will be avoided” (p.35), despite its frequent and simple explanation by others, and to which he attaches great importance for its “arithmetization of syntax” (same page). Therein lies the objectionable connection of the theorem with mathematics.”Arithmetization of syntax” discloses that “Goedel numbering” is performed on linguistic components of the arguments. That is to say, these components, from single letters to series of sentences are each assigned numbers, in impressively intricate arrangements, and then it is said the contents of those arguments somehow apply to mathematics. It is not recognized that the numbering merely concerns the language in which the arguments are couched, not the contents of that language.To support their reasoning, the proponents offer all kinds of analogies. In this book the author uses (p.36) the comparison of binary data, the mathematical collection of bits 0 and 1, as representing sounds and pictures in computer games. Here, however, the subject is physics, to which mathematics applies, even if not observable in the result. Similarly there are other examples in which the subject matter has some connection with mathematics. But the subject matter in Goedel’s arguments, the content of the “Goedel sentence” and of the logic applied, does, again, not concern mathematics. Only the language, by being designated with numbers, does.It should be appropriate here to go back to that Goedel sentence and the associated logic. I discussed these in other reviews, and I might now first provide a simple form of that sentence again, looked for in vain in the book reviewed. It is:THIS STATEMENT IS UNPROVABLE (IN THE SYSTEM).As noted previously, Goedel’s alleged proof of this statement is said derived from outside the system and accordingly not to be contradicting the statement. But I pointed out that the rules of logic used can be incorporated into any system, a procedure that should be allowed in order to find what is or isn’t logically possible, and therefore the proof would indeed be a contradiction. I noted in fact another contradiction resulting via simple logic: THE STATEMENT CAN BE PROVED UNPROVABLE, since if provable it would be contradicted; AND THE STATEMENT CANNOT BE PROVED UNPROVABLE, since if proved it would again be contradicted.What is significant is that the statement, thus harboring contradictions, cannot be added to the axioms of the system as suggested by the discussants, because that would make the system inconsistent, with consistency vehemently, and justly, insisted on by all authors. The discussants believe that the statement is legitimate, because it is a “well formed formula”, i.e. it abides by grammatical rules. But contradictions are possible within the best of grammar. The problem is better attributed to positing “formal systems”, ones without meaning, since in those cases one has no content to fall back on for the search of hidden contradictions.Hidden contradictions are the province of paradoxes, to which the Goedel sentence can be relegated, and which I also consider, but elsewhere:

⭐.

⭐I have recently become fascinated by Gödel’s Incompleteness Theorem, although I am struggling since I don’t have an aptitude for maths. I’ve read the classic “Gödel’s Proof” by Nagel and Newman, and the chapter on Godel in Ian Stewart’s “Concepts of Modern Mathematics”, and I’ve also read the chapter in David Papineau’s “Philosophical Devices”. This book provides a useful, discursive overview of the Theorem which I found helpful since it is simpler than the first two but more detailed than the Papineau. The latter is the best introduction, but I would recommend reading this present volume next before going on to anything more detailed. What I found most interesting is the author’s emphasis that Godel’s Theorems only apply to formal systems containing a certain degree of mathematics. This important qualification is a useful correction to some of the wilder expositions on the internet, where people use their misconception of what Godel actually saying as licence to justify any old crackpot theory, on the grounds that “all theories are unprovable”.

⭐I found this book clarifying on many uses of Godel’s Theorem and on mathematics or science in general. Torkel Franzen’s insights were sobering and demystifying. I found myself shedding a few preconceptions and misconceptions.However, this book is definitely NOT for persons without a mathematical background. I agree with other reviewers on this one.It felt like when Torkel Franzen was writing he was skipping steps in his thought process which to him as a mathematician were trivial but to the reader are not. I have a Computer Science degree but I had to re-read many paragraphs to figure out how and why the author was reaching his conclusion.

⭐Viele völlig absurde Fehlinterpretationen über die Unvollständigkeitssätze nach Gödel existieren. Jeder mathematisch interessierte Mensch, der herausfinden möchte, worum es wirklich bei Gödels Sätzen geht, sollte bei der Lektüre dieses Buches auf seine Kosten kommen. Kurz gesagt: Eine schöne kleine Abendbeschäftigung, die mich ab und an zum Schmunzeln und viel zum Nachdenken brachte.

⭐

⭐Es una obra interesante y bastante clara, lo cual no es frecuente en obras de esta naturaleza.También trata de lo mucho que se abusa de estos famosos resultados de Gödel.

⭐

⭐Nice book, though a bit biased treatment

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