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Gödel's Theorem: An Incomplete Guide to Its Use and Abuse (Englisch) Taschenbuch – 30. Juni 2005

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" "Franzen's book is accessible, well written, and often funny..." -Richard Zach, History and Philosophy of Logic, July 2005 "Ich mochte allen meinen Lesern ... ein Buch ans Herz legen, und zwar "das Neue" von Torkel Franzen: Godel'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 Godel'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 Godel'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 Godel'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 GAodel 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 Godel'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, Godel'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"

Synopsis

The theorem is tossed about and misapplied by the uninformed, so the author gathered up quotes and responses he's been involved with on the internet and tackled a presentation for what he terms a "general audience." Mathematicians will turn to more sophisticated treatments; determined non-mathematicians with a strong bent for formal logic will be a

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Few theorems, of pure mathematics have attracted much attention outside the field of mathematics itself. Lesen Sie die erste Seite
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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.
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Gödels Unvollständigkeitssätze waren eine Revolution in der Mathematik, und es gab Missverständnisse und Auswirkungen, die in anderen Bereichen kräuselte sich. Der erste Hauptsatz in einer österreichischen Fachzeitschrift im Jahre 1931 erschien und sehr einfach ausgedrückt werden.

In jedem konsistenten formalen Systems S, in dem es möglich ist, eine Mindestmenge an die elementare Arithmetik durchzuführen, gibt es Aussagen, die weder bewiesen noch widerlegt werden kann.

Die wichtige Konsequenzen sind enorm, fatto che es bedeutet Clustering, die in jedem System, das verwendet werden kann, um Rechenaufgaben wird es Theoreme, niemals als wahr oder falsch überprüft werden können. Mit anderen Worten, werden einige Kenntnisse immer unerreichbar innerhalb dieses Systems sein. Natürlich schließt dies nicht aus, wobei Sie zusätzliche Axiome, mit denen andere Theoreme zu beweisen war.

Franzen hat eine ausgezeichnete Arbeit bei der Erklärung der Unvollständigkeitssätze in einer Weise, die von Menschen mit einer begrenzten Kenntnisse der Mathematik verstanden werden kann. Zwar gibt es einige Stellen, an denen ein High-School Mathematikunterricht ist nicht ausreichend, um durch weitere technische Argument zu verstehen, genügt es, die Sätze zu verstehen und zu schätzen wissen.

Meine Lieblings-Teile des Buches waren die Abschnitte gewidmet "Anwendungen" des Unvollständigkeitssatzes außerhalb der Mathematik. Einige Beispiele sind von der Religion, Politikwissenschaft und Philosophie.
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Amazon.com: HASH(0x98977e04) von 5 Sternen 24 Rezensionen
112 von 115 Kunden fanden die folgende Rezension hilfreich
HASH(0x989a4e58) von 5 Sternen Knowing when to invoke Gödel's Incompleteness theorems 7. August 2005
Von T. Gwinn - Veröffentlicht auf Amazon.com
Format: Taschenbuch
As evidenced from the title, the primary focus of the book is to identify the specific nature of these theorems, where they apply directly, and where they do not apply directly, and where they are interpreted entirely erroneously.

Although the book is aimed at non-mathematicians and those with no knowledge of formal logic, I can't really imagine someone with no understanding of logic and some fair amount of math comprehension benefitting alot from this book. I mean, by p. 10 he talking about Diophantine equations and Goldbach-like conjectures, and soon after, "PA" and "ZFC" are tossed about as if they were practically everyday acronyms for most people. The book is however, largely free of formulas and proofs, for those who are dissuaded by such. The overviews of the theorems themselves is not as lucid as I imagine they could be (which is why I rate it a 4 instead of a 5). The overviews will also seem a bit alien to someone expecting and Nagel & Newman kind of treatment; instead, this is discussed from a more abstract perspective of the characteristics and properties of formal systems, which avoids getting into the gritty details (even Gödel-numbering is not explained in detail!) but may be hard to grasp for someone not used to thinking at this level of abstraction about mathematical systems.

With that said, I still think it is quite worthwhile reading, and at a slim 170ish pages, it is a fairly quick read. After the overviews, he takes on various applications/misapplications of the theorems by topic. So, there are discussion of the theorems' relevance or applicability to things such as TOE (Theory of Everything), Turing machines, skepticism, minds, inexhaustibility, computability and so on. He does so typically by first giving several quotes that appear either in the literature or commonly on the internet, and then proceeds to either correct or clarify those quotes. Such notables as Roger Penrose, Freeman Dyson, and Stephen Hawking are among the quoted who are scrutinized.

I think the primary goal of the book is accomplished in its debunking of outright misuses of the theorems, and by way of correction and clarification of other uses, it accomplishes its pedagogical goal. I know it will cause me to strive to be more precise in any future invocations of these theorems.
49 von 50 Kunden fanden die folgende Rezension hilfreich
HASH(0x989a4fa8) von 5 Sternen Just about the best book on Godel's Theorem I have read 6. April 2006
Von sigfpe - Veröffentlicht auf Amazon.com
Format: Taschenbuch
As a mathematician I thought I had a good understanding of Godel's Theorem but Franzen highlighted a bunch of misconceptions that I had. It went on to answer a bunch of questions I had wondered about but had never had a chance to ask a logician. So in a sense this book is precisely what I wanted from a book on Godel's theorem and I can't help but give it 5 stars.

But that's not all. Over the years I have read countless papers, articles and books by author who invoke Godel's Theorem in the most inapposite paces without understanding it. I've found this to be pretty annoying and in many cases there is no rebuttal. Franzen tackles many of these misuses whether they are comments on USENET or published arguments by Lucas, Chaitin and Penrose. It's great to see someone put pen to paper and reply to these abuses in one place. That would bump my rating up to 6 stars if I were able.

But be warned, this book is challenging. I'd suggest that as a prerequisite you need to be a mathematician, philosopher or computer scientist with at least some familiarity with Godel's Theorem.
31 von 31 Kunden fanden die folgende Rezension hilfreich
HASH(0x989a5300) von 5 Sternen Valuable Counterbalance to Widespread Misconceptions and Nonsense in Print and on the Internet 13. August 2006
Von Michael Wischmeyer - Veröffentlicht auf Amazon.com
Format: Taschenbuch
Torkel Franzen has created an immensely valuable, deeply fascinating examination of misunderstandings, misconceptions, and outright abuse of Godel's theorems frequently found on the Internet (and occasionally in print). He does so in a cogent, non-confrontational style that makes enjoyable reading. Godel's Theorem - An Incomplete Guide to Its Use and Abuse warrants five stars.

A word of caution is appropriate, however. Chapters 2 and 3 will be heavy going for readers not familiar with formal logic. Although Franzen avoids the details of Godel numbers in his explication of Godel's proof, he does delve into topics like self-referential arithmetical statements, Tarski's theorem, Rosser sentences, weaker variants of the first incompleteness theorem, computably decidable sets, Turing's proof of the undecidable theorem, and the MRDP theorem.

Furthermore, the appendix offers both a formal definition of the concept of a Goldbach-like arithmetical statement and comments on the significance of Rosser's strengthening of Godel's first incompleteness theorem. (Any reader that stays the course with the early chapters will be able to handle the appendix discussions. The short chapter 7 is also more technical as it discusses the completeness of first order logic.)

A word of encouragement is equally appropriate. Chapters 2 and 3 can be browsed, even skipped outright. The later chapters are much more accessible and don't require that the earlier chapters have been mastered; instead, they focus on examples of the misuse of Godel's theorems - from the merely technically inaccurate to the humorously nonsensical. It is these later chapters that makes this book special.

Although words like consistent, inconsistent, complete, incomplete, and system have been carefully defined within the context of formal logic, in normal discourse these words have varied meanings, often leading to vagueness and confusion in discussions of Godel's theorems. Furthermore, Godel's theorems often serve in an inspirational fashion, that is being used as analogies and metaphors in which the essential condition that a system must be capable of formalizing a certain amount of arithmetic is largely ignored.

Invocations of Godel's incompleteness theorems in theology, in physics (like the theory of everything), and in the philosophy of the mind (the Lucas-Penrose arguments) are found in chapters 4, 5, and 6. Chapter 8 addresses the widely publicized philosophical claims of Geoffrey Chaitin on the relationship between incompleteness and complexity, randomness, and infinity.

Godel's Theorem - An Incomplete Guide to Its Use and Abuse may be too much too soon. A reader new to Godel's work might consider starting with Godel's Proof (by Ernest Nagel and James R. Newman) or Incompleteness - The Proof and Paradox of Kurt Godel (by Rebecca Goldstein).
17 von 18 Kunden fanden die folgende Rezension hilfreich
HASH(0x989a4e10) von 5 Sternen Gödel's Theorem 16. Februar 2011
Von Sam Adams - Veröffentlicht auf Amazon.com
Format: Taschenbuch
This book seems cobbled together, and its exposition is unclear. Chapter one is a short seven-page introduction. Chapter two, entitled "The Incompleteness Theorem: An Overview", is forty-eight pages long, and this is where beginners are going to get frustrated and discouraged. Franzen does not sufficiently clarify for the uninitiated the difference between mathematical logic (first-order logic) as an axiomatic system and a first-order theory (which he usually calls a formal system). So he also doesn't clarify (until he finally does, in chapter seven) the difference between the completeness of first-order logic and the incompleteness of the first-order theory of arithmetic (his mention of negation completeness isn't clear enough to be helpful). Nor is he helpful on the concepts of true or truth in relation to logical truth (tautology), to axiomatic systems, to consistency, to human knowledge, to number theory, or to mathematics. He does not speak directly upon the distinction between validity and provability. He isn't explicit about the distinction between well-formed formula (statement, sentence) and theorem. Confusion about these distinctions leads to confusion about the meaning and range of applicability of Gödel's Theorems, so Franzen has not done a good enough job in this overview chapter of educating the uninitiated.

In chapter three, Franzen attempts to introduce the notions of a computably enumerable set and a computably decidable set. He seems to believe that because he knows what he's talking about that what he's saying is clear. He obscures the presentation with the clutter of inelegant examples and talking about what a computer can or cannot theoretically do rather than sticking with the mathematical concept of formal computation and speaking in that context about algorithms. On the complaint about his examples, simply using the standard 26 lower-case letter alphabet and discussing strings of letters would have sufficed; he could have left out the vowels if he wanted to avoid the possible confusion arising from "words" in a list. On the complaint about his referring to computers, he's obscuring the point that Gödel's proof is a mathematical proof and the concepts used in it, such as computability, are mathematical concepts (by the way, he never mentions Church's Thesis in the book). Later discussion about AI, Gödel's Theorem, and the human mind is soon enough to bring in the digital computer.

This chapter, in fact, seems out of place. Franzen informally discusses computably enumerable sets and computably decidable sets, then states three theorems linking the concepts:
[1] "Every computably decidable set is computably enumerable."
[2] "A set E is computably decidable if and only if both E and its complement are computably enumerable."
[3] "There are computably enumerable sets that are not computably decidable."

He continues with a sketch of Turing's proof of [3], then builds up to using a celebrated theorem by Matiyasvich, Robinson, Davis, and Putnam concerning diophantine equations to derive incompleteness using methods unlike Gödel's and which do not employ any sort of self-reference. But this isn't what most readers, who are curious how Gödel's proof works and how it applies outside of formal number theory, are interested in learning at this point in the book. Discussion about alternate proofs of incompleteness belong at the end of the book, after having satisfied the reader's curiosity about Gödel's particular method.

In fact, nowhere in this book (including the appendix) does Franzen provide the kind of detail on Gödel's method as is available in the seventh chapter of Nagel and Newman's book Gödel's Proof. Whatever the reader gleaned from the patchwork of chapter two is all there is to get. Franzen discusses a second alternative proof of incompleteness, based upon Kolmogorov complexity, in chapter eight. Since misapplications of Gödel's Theorem are based on misunderstanding Gödel's actual method of proof, not merely on the mathematical meaning of what he proved, it makes no pedagogic sense to avoid discussing how Gödel achieved what he did.

In chapter four, Franzen moves on and explains in general terms, with only a momentary mathematical hiccup, why various applications of Gödel's Incompleteness Theorem are misguided. Chapter four isn't very interesting, however, because the examples he uses are so clearly misapplications. The obvious refutation is simply to point out that Gödel's Theorem is concerned with a very specific kind of limitation of certain formalized, axiomatic, mathematical theories and the example isn't one of those.

Chapter five concerns the second part of Gödel's Theorem which proves that the formal system Gödel examined cannot contain a proof of its own consistency. Essentially, Franzen makes a distinction between wanting a proof (such as of consistency) in order to be convinced that something is true and wanting a proof (such as of consistency) in order to be shown that something, of which you may already have the conviction that it is true, that it is true. But he doesn't make the distinction clear and distinct from the start.

In chapter six, Franzen takes on Lucas and Penrose. His main argument is that we do not know, of an undecidable statement U of formal system S, whether it or its negation is true because we only know that within S it is derivable that if S is consistent then U is derivable, but we do not know that S is consistent (because if we did, then by modus ponens we could derive U, which by the Incompleteness Theorem we know is not derivable in S, unless S is inconsistent), and therefore it can only be a belief that U is true, founded on the belief that S is consistent, unless the consistency of S has been proven in a formal system other than S, in which case we know that S is consistent and thereby that U is derivable and hence true. Franzen is blurring the concept of knowledge. He is relying on his distinction from the previous chapter on proof for conviction of a truth versus proof for exhibition of a truth. One is epistemological, the other is purely formal.

Chapter seven concerns the completeness of first-order logic, and Franzen gives the reader a quick glance at a few concepts needed to understand the difference between this kind of completeness and the kind to which Gödel's Incompleteness Theorem refers. He informally states the completeness theorem as (I'll change his italic A to 'Q' and slightly edit his sentence): "If Q is a logical consequence of a set of axioms, then there is a proof of Q using those axioms and the logical rules of reasoning." He combines the soundness theorem (which he also informally states and which uses the notions of logical consequence and model) and the completeness theorem into a theorem which he states in two logically equivalent forms for a first-order theory T (I'll change his italic A to 'Q'): "A sentence Q is true in every model of T if and only if Q is a theorem of T." (and) "T has a model if and only if T is consistent." After this he gives a set of axioms for peano arithmetic (the PA to which he's been referring throughout the entire book) and talks briefly about non-standard models of PA. This chapter should have occurred in modified form much earlier. As I remarked at the beginning, this book seems cobbled together.

The final chapter (besides the appendix) discusses Kolmogorov complexity and Chaitin's version of the Incompleteness Theorem. Informally, the complexity of a string of symbols is measured by the length of the shortest algorithm that produces that string. Chaitin's version of incompleteness states that: For a consistent formal system T (satisfying certain arithmetical conditions), there exists "a number c depending on T such that T does not prove any statement of the form 'the complexity of the string s is greater than c'." Because there are in fact strings of greater complexity than c, it follows that T is incomplete, "unless it [that is, T] proves false statements about complexity". Franzen remarks: "This theorem ... doesn't say anything about the complexity of the theorems of a theory, but instead deals with theorems that are statements about complexity." Franzen spends more time on this idea of complexity and Chaitin's claims about the implications of his proof than he ever does on the details of Gödel's work. He concludes this chapter with a discussion (having nothing to do with complexity or Chaitin's ideas) of set theory and axioms of infinity. This chapter is the most carefully written in the book, but also the farthest from what most people would be reading the book to learn.

What you don't get in the appendix is a look at how Gödel proved his Theorem. Instead, Franzen rushes through a set-up of PA using logical notation, waves his hands about the applicability of the alternative incompleteness proof discussed in chapter three, talks at a rapid clip about Rosser's proof and Robinson arithmetic, then jumps into remarks about bounded formulas, the algorithmic decidability of their truth, and sets of natural numbers defined by bounded formulas being computable sets, yet conversely there being computable sets not defined by any bounded formula. He uses all of this to reformulate and tighten the meaning of his term 'Goldbach-like sentence' that he introduced in chapter two and used on and off throughout the book.
17 von 18 Kunden fanden die folgende Rezension hilfreich
HASH(0x989a57bc) von 5 Sternen An Excellent Book Clarifying the Proper Use (And Abuse) of Godel's Theorems. 13. Januar 2007
Von New Age of Barbarism - Veröffentlicht auf Amazon.com
Format: Taschenbuch
Ever since the logician Kurt Godel (1906-1978) proposed his famous Incompleteness Theorems, various claims have been made about their implications. In particular, it is largely agreed that the Incompleteness Theorems were revolutionary in the fields of mathematics (as well as philosophy, computer science, and science in general). However, the "revolutionary" nature of these theorems is highly exaggerated (as the author explains in this book). Indeed, the theorems are frequently used to justify all manner of assertions about what can be known or proven (e.g. the theorems are even used to explicate Zen Buddhism, with its koans, among other things). In particular, the theorems have been claimed to justify various popular claims made in the philosophy of mathematics, science (particularly regarding the so-called Theory of Everything (TOE) hoped for in theoretical physics), religion (including theology), and mind, as well as being used as arguments for skepticism regarding truth and for postmodernist philosophies. In the book _Godel's Theorem: An Incomplete Guide to Its Use and Abuse_, Torkel Franzen contends with some of these assertions showing how many of them are highly problematic. Franzen (who provides plenty of examples of the use of such arguments from his discussions on the internet) attempts to delineate exactly what is and is not implied by these theorems. Franzen contends that many of the claims made (allegedly supported by the proof) are simply erroneous, but that others can be understood as relying on analogy (appealing to the idea of self-reference and using Godel's Theorems as a sort of inspiration). Franzen does not denigrate the use of analogy in such cases but merely points out the fact that analogy is limited and that frequently deeper justification is called for. In this sense, the book is extremely helpful in that it shows exactly what can and cannot be derived from the theorems, as well as many of the errors and assumptions made in the various claims. However, it should be noted from the beginning that this book is extremely difficult. Despite what the author says in the preface, this book is probably not for those who do not have a background in mathematics or philosophy and who are not used to the methods of mathematical proof. Further, many of the arguments presented in this book are extremely subtle (and even after carefully reading through it, I still cannot be sure that I have grasped all of them or that I can make the proper distinctions). Thus, even for the advanced reader the book needs to be read with care. In fact, as Franzen effectively shows even among many famous and well-respected scientists, mathematicians, logicians, and philosophers (those who should know better!), the implications of the theorems are not widely understood.

In his famous address to an international congress of mathematicians in 1900, David Hilbert made his famous appeal to mathematicians calling for mathematical optimism ("non ignorabimus") regarding the prospects of mathematical proof. It is widely believed, however, that by proving his Incompleteness Theorem, Godel effectively demolished Hilbert's program and refuted optimism. There are actually two incompleteness theorems of Godel (extended by Rosser so as to include a stronger notion of consistency and not merely "omega-consistency"). They are as follows:

First Incompleteness Theorem: Any consistent formal system S within which a certain amount of elementary arithmetic can be carried out is incomplete with regard to statements of elementary arithmetic: there are statements which can be neither proved, nor disproved in S.

Second Incompleteness Theorem: For any consistent formal system S within which a certain amount of elementary arithmetic can be carried out, the consistency of S cannot be proved within S itself.

In Chapter 2 of this book, Franzen explains fully these two theorems, defining a formal system (two formal systems that will play a role in this discussion are that of Peano Arithmetic (PA) and the Zermelo-Frankel axioms of set theory (with the axiom of choice) (ZFC) in which normal mathematics is conducted), as well as what it means for that system to be consistent and complete. (Franzen defines what he terms "Goldbach-like statements" in his definition of consistency and soundness.) Franzen also discusses what is meant by "a certain amount of elementary arithmetic" (and also brings up a common misunderstanding of Godel's original Completeness Theorem for first-order predicate logic). Franzen dismisses a common misunderstanding that these theorems say something about "complexity" of a formal system. Franzen also shows how formal systems relate to the theory of computability (developed by Turing). In addition, Franzen argues that while Godel's Theorems may appear to refute Hilbert's optimistic claims, that it is non-obvious that they would apply to the undecidability of any questions that occur in normal mathematics (such as the Goldbach conjecture; although later he shows how they have been shown to apply to certain obscure combinatorial questions). The proof of Godel's Theorems involves making use of Godel numbers assigned to sentences in the language and then making use of a self-referential condition which amounts to the so-called Liar's Paradox (i.e. "This sentence is false."). Franzen then proceeds to show how various attempts to justify postmodernism based on these theorems in fact rest on a misunderstanding (in that mathematics does not "branch off" as suggested by the postmodernist). Franzen also explains exactly what Godel believed his theorems said about the human mind (and this differs from some of the more radical attempts to argue that the mind is non-mechanistic based on the theorems). In Chapter 3, Franzen considers computability, formal systems, and enumerability. Franzen explains how computability relates to formal systems and defines the notions of enumerable and decidable for sets of strings. Franzen then proves the theorems using these methods (as shown by Turing). There are however various tricky issues involved here and a careful reading of this chapter is required. In Chapter 4, Godel considers some of the implications of the theorems for philosophy. Here, he shows that the theorems say nothing about formal systems which may not include arithmetic (such as the Bible or Ayn Rand's philosophy, etc. considered as such). Franzen also shows what these theorems do and do not have to say about "human thought". Franzen also considers so-called "generalized Godel sentences" (in both mathematical and non-mathematical contexts). Franzen also considers various arguments put forward in physics against a TOE (such as by Hawking and Dyson) making use of these theorems and shows how these arguments are not valid. Franzen also considers various theological arguments relying on these theorems as a justification for faith (or for atheism as the case may be), and shows how such attempts are also not valid. In Chapter 5, Franzen considers the case for skepticism made with the help of these arguments. Franzen shows how the arguments do not in fact support a case for mathematical skepticism. Franzen also explains exactly what is meant by "mathematical inexhaustibility" and how this relates to the case for skepticism. (Again, the argument here is extremely subtle and interesting, and this chapter should be read very carefully.) In Chapter 6, Franzen considers the question of what the theorems have to say about minds and computers. Franzen discusses an idea of Rudy Rucker regarding a "Universal Truth Machine" and shows how this rests on a false understanding. Franzen also considers arguments put forward by Lucas and Roger Penrose (and shows how some of them are justified but how others are problematic). Franzen also considers mathematical inexhaubibility again as well as the ability to "understand one's own mind" (referencing an analogy of Hofstadter's). However, it should be noted that Franzen's analogy to the systems PA and ZFC regarding self-understanding is itself nothing more than an analogy, and thus suffers from the same problems as the analogy of Hofstadter's regarding the inability to attain self-understanding. In Chapter 7, Franzen considers the question of Godel's Completeness Theorem, showing a common confusion that arises from this theorem and the Incompleteness Theorems. In Chapter 8, Franzen provides a very interesting discussion of incompleteness, complexity, and infinity. Franzen illustrates Chaitin's Incompleteness Theorem (which relies on a notion of complexity and a form of Berry's paradox). Franzen also shows how some of Chaitin's claims about randomness may be problematic. Finally, Franzen considers various questions concerning infinity; particularly the Continuum Hypothesis (CH) and nonstandard models of arithmetic (Robinson). The book ends with an appendix which provides a more detailed exposition of the "Goldbach-like statements".

This book is very interesting and useful; in that, it provides an excellent clarification of the role of Godel's theorems. These theorems are frequently abused by philosophers to make points which they do not in fact make. The arguments in this book are extremely subtle and may be difficult to follow; however, I believe that fundamentally they are sound and thus provide an excellent understanding of exactly where an appeal to the Incompleteness Theorems is and is not justified.
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