Gödel's Theorem: An Incomplete Guide to Its Use and Abuse

by Torkel Franzén and Torkel Franzén

Book Overview

"Among the many expositions of G?del's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franz?n 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...

• Format: Paperback
• Language: English
• ISBN: 1568812388
• ISBN13: 9781568812380
• Release Date: June 2005
• Publisher: A K PETERS
• Length: 182 Pages
• Weight: 0.55 lbs.
• Dimensions: 0.4" x 6.3" x 9.0"

Customer Reviews

An excellent choice for courses that cover the philosophy of science and mathematics

Godel's Incompleteness Theorems were a revolution in mathematics and there were repercussions and misunderstandings that rippled out into other fields. The main theorem first appeared in an Austrian journal in 1931 and can be stated very simply. In any consistent formal system S within which it is possible to perform a minimum amount of elementary arithmetic, there are statements that can neither be proved nor disproved. The consequences are enormous, in that it means that in any system that can be used to perform arithmetic, there will be theorems that can never be verified as either true or false. In other words, some knowledge will forever be unattainable within that system. Of course, this does not preclude adding additional axioms that will allow other theorems to be proved. Franzen does an excellent job in explaining the incompleteness theorems in a manner that can be understood by people with a limited knowledge of mathematics. While there are few places where a high school mathematics education is not sufficient to understand a more technical argument, it will be enough to understand and appreciate the theorems. My favorite parts of the book were the sections devoted to "applications" of the incompleteness theorem outside of mathematics. Some examples are from religion, political science and philosophy. Godel's theorems are used to "prove" that no religion can contain a complete set of answers and that any constitution must of necessity be incomplete. Human thought is also interpreted in the context of the incompleteness theorems. The statement is: Insofar as humans attempt to be logical, their thoughts form a formal system and are necessarily bound by Godel's theorem. This statement and others related to the nature of human thought are examined in detail. The philosophy of Ayn Rand is also examined as a system that must of necessity be incomplete. This book would be an excellent supplemental text for a philosophy course where the nature of truth is examined. It would also be a very good choice for a course in the philosophy of mathematics. Published in Journal of Recreational Mathematics, reprinted with permission.

An Excellent Book Clarifying the Proper Use (And Abuse) of Godel's Theorems.

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 i

Valuable Counterbalance to Widespread Misconceptions and Nonsense in Print and on the Internet

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).

Just about the best book on Godel's Theorem I have read

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.

The Absolute Best Semi-Popular Book on Godel's Theorems

First, this book is amazingly beautifully written. Franzen sets a high standard for writing that we should all aspire to. As to content, there are two distinct topics which are interrelated in the book: one is the abuse to which Godel's Theorems--there are two of them--are often put in popular and sometimes even technical writings. This part is very enjoyable to read for its clear explanations and its low-key sense of humor. The second topic, related to the first, is an extremely succinct, well-written exposition of Godel's Theorems. Franzen's careful exposition is quite illuminating even to those who thought they understood these theorems. In addition, his detailed account shows that many alternative accounts are either misleading or flat-out wrong. Most of his explanations are easily grasped by anyone who reads them with an ounce of care. More technical aspects are included in appendices. None of the other works covering similar material can hold a candle to this terrific book (which also includes the earlier popular text by Nagel and Newman, which is not well-written and contains some mistakes as well). Anyone with even a passing interest in Godel's Theorems (and how they are often misused) should purchase and read this book. It is certain to become the definitive work on the subject.