Gödel's Proof

When growing up I got my first introduction to Godel from this little book, and it did serve the purpose of whetting my appetite for logical subjects; still there was plenty of it that seemed a little fuzzy to me back then. Having fully wrapped my brain around the proof in college (while working on my math degree), I've found it interesting recently to go back and give Nagel and Newman another look. My feeling today is that my earlier confusion in certain passages was not pure naivete; the authors' notation and their explanations were often to blame. Hofstadter's improvements help a bit, but I think basically a fresh start is needed if we want a truly accessible look at Godel.Some specific, technical examples follow. (1) Nagel and Newman retain Godel's original scheme for numbering formulas by using prime powers; while this is not a huge intellectual obstacle, more intuitively accessible options exist, such as an ASCII-like scheme (Goldstein's otherwise garbled account has the advantage that it pursues this course). Certainly Godel's system results in some very intimidating-looking formulas when you try to spell out an example or two. (2) When presenting the crucial self-referential Godel sentence, Nagel and Newman use the following notation: ~(Ex) Dem(x, Sub(y,17,y)) (Hofstadter added the capital letters, which represent a slight improvement). While this is fairly close to Godel's original notation, it is again _way_ less accessible than it might have been. All the following points contribute to the attentive reader's confusion: special mental effort is required to figure out the purpose of the number "17"; the two instances of the letter "y" are being used in totally different ways; even though this is supposed to represent a sentence in the formal language ("PM"), it is not at all clear how the "function" Sub is supposed to fit in with the formalism as previously presented. Once again, if N and N had thought a little more about their choice of notation, all these hindrances to comfortable understanding could have been avoided. (3) When N & N incorporate analogies into the text, in an effort to increase user-friendliness, the results are often more confusing than helpful. An odd little vignette involving taking numbers at the deli counter requires considerable effort to make sense of; the payoff in terms of insight is minimal. The authors devote a fair amount of time to the Richard paradox, which itself requires a fair amount of toil to understand, but which in the end is only a modestly illuminating analogue to Godel's sentence; they would have done better to put Richard into a footnote, and let us concentrate on the topics that deserve our attention.Finally, as Hofstadter points out in his introduction, N & N close with some highly dubious philosophical remarks, concerning matters like the relative strengths of human thinking and "calculating machines" (a typical antiquated bit of terminology). Although Hofstadter doesn't diagnose it this way, I think the root of the trouble lies in the way they depict the relative abilities of formalists and metamathematicians. Readers are left with an image of the former trapped within the confines of their formal system and at a loss to say anything about Godel's sentence, while on the other hand metamathematicians with uncanny powers hand down pronouncements about the sentence’s truth. The fact is that not even metamathematicians can really prove Godel's sentence, only the fact that the consistency of the system implies this sentence; but as it happens, this conditional statement is one that formalists are also perfectly capable of proving (as Godel showed in the run-up to his second theorem). This observation, to my mind, undermines the whole self-serving narrative of savvy, creative humans out-proving those mindless formalists and "calculating machines."

We are in 1931, some clever people like Hilbert, Russell and Whitehead have been trying to put mathematics on solid foundations. But a young guy called Gödel just constructed a proof stating that all consistent systems are incomplete. Then if mathematics has no contradictions (consistent), it cannot be composed by theorems (incomplete). Therefore, it is impossible to validate mathematics using mathematics. Damn, this kid just shocked the whole scientific community. How on earth someone proof that ALL consistent systems within the whole universe of possible systems are incomplete? This book will give you some clues. If you want to see and hear it, read GEB by Douglas Hofstadter.

Overall, I very much enjoyed this book. It was as promised, a complete, though not thorough explanation of Godel's incompleteness theorem.While it did use mathematical syntax throughout the book, it was all in the context of what was necessary for the explanation (and in many cases the mathematical symbols are easier to parse than the written explanation, however usually both are provided). Furthermore, tables are often shown throughout the book to further reinforce concepts.As someone with an engineering rather than formal mathematics background, I found the book overall quite easy to follow. That is not to say that there weren't a few pages for which I needed to re-read several times, or that I would jump backwards in the book to reacquire the context of what was being explained, but it seemed like the book chose the right trade offs overall.It is worth noting that the book is quite a quick read. I imagine most people would get through it in a little over a couple of hours. Yet the length of the book feels right. It manages to gloss over the right areas, while not affecting the subsequent explanations negatively.

Personally, I think the author did good job as he has tried to explain this very complex topic in relatively simple words. However, this book is intented for somebody already acquainted with mathematics, mainly with basic of mathematical (or formal) logic. The author describes mainly the topic itself, only sometimes he deals with historical backdrop. So, I would not recommend this book to somebody interested in history of science.The book is divided to two main parts. In the first one (chapters 1 to 6) the author brings introduction to terms like a "consistency" and "completeness" in logic and he desribes methods how to prove that some formal system is consistent or complete. The author is also talking about some logical paradoxes. Then some necessary information about mapping in maths is provided. The second part of the book (namely chapter 7) contains Godel's proof itself. Since the proof is not very simple, the author fisrtly introduces some other auxiliary theorems. Finnaly, the proof is coined and explained in "natural" language. I highly appreciated many footnotes sheding more light on, at first glance, not very clear terms and statements.To sum up, the book is worth reading and I think that everybody either using advanced maths in day-to-day working or dealing with computer sciences should read this book.

I have to go against the flow of reviews and give this 3. I am not a mathematician, but a computer scientist (who has worked with applied mathematics). I found Nagel and Newman's exposition of Gödel's theorems good to a point, but, despite several re-reads, the "click" of the elusive intuition didn't materialise.I continued my search to understand Gödel and found a more intuiitive exposition in "Incompleteness" by Rebecca Goldstein. The key difference in Goldstein's exposition was to make clear the "diagonal lemma". Goldstein states that Gödel "didn't actually use it, but rather derived the particular case"; it does however convey the entrance of self-referentiality more clearly. Nagel and Newman's text does not mention the "diagonal lemma". I would recommend Goldstein's text to lay persons seeking an intuitive grasp of the incompleteness theorem.

This is quite helpful if you are going to tackle Godel's key text 'On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems'

A clear and accessible intro to the problems of Mathematical completeness and formalism. I was especially impressed with the description of the Godelian system for numbering characters, strings and lists of strings.

Great well paced tutorial however it will exercise your logic skills. This book is worth the effort though. Fast delivery and in good condition

This is a very accessible introduction to a very difficult proof.

Achei este livro EXCELENTE porque ele mostra como a lógica está acima do próprio sistema axiomático da matemática. O livro é indicado tanto para matemáticos como para não matemáticos. Gostei da maneira como os autores explicam cada argumento apresentado. Este é o livro mais didático sobre o teorema da incompletude de Gödel.

Excelente.Una verdadera introducción a la prueba más importante de la ciencia abstracta hasta ahora.El libro ofrece una adecuada introducción, llena de ejemplos y diversos elementos que permiten la comprensión del esquema de la prueba.

I bought this based on praise from Doug Hofstadter. It helped him understand Godel's theorem. If you're already familiar with the intuition of Godel's theorem and not able or willing to actually read his proof, this book is a great middle ground between the two levels of understanding.I also discovered it is an excellent sleep aid. Through use of this text I am able to fall asleep within minutes at virtually any time of day.

Although some details of the proof are missing, and some theorems just given without further details, the most fundamental aspects are explained in the text, and some details given in the appendix. With the "vocabulary", grammar and logic principles taught by the book it is even sometimes possible to find the deduction ("proof") by just experimenting with what you have learned. Additional literature is not necessary, but after reading this book highly recommended, as it is a good foundation to seek further understanding.For those who like think about the connection to other discplines of science, "Goedel, Escher, Bach" is also recommended - although you may find some redundancy; which might also mirror the special nature of the given topic.

Leí este libro después de leer en G.E.B que este pequeño libro inspiró a Hofstadter a escribir GEB.Por sí mismo, este libro es una muy buena introducción a los Teoremas de Gödel, y creo que me impresionó aun más Hofstadter al ver lo que salió después de ser inspirado por este libro.

Um texto clássico sobre o famoso Lema de Gödel, editado pela Imprensa Universitária da Universidade de Nova Iorque. Trata de forma didática sobre as publicações feitas por Kurt Gödel na década de 1930.

Si vous vous intéressez au mathématiques et à la logique, la lecture de cet ouvrage, qui présente de manière (relativement) simple la démonstration du théorème d'incomplétude de Gödel, vous apportera beaucoup de satisfaction.

Excelente.

Well written

Great book for those interested in the limits of mathematical thinking.