Gödel’s Proof by Ernest Nagel (PDF)

Description

An accessible explanation of Kurt Gödel’s groundbreaking work in mathematical logicIn 1931 Kurt Gödel published his fundamental paper, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as “one of the greatest contributions to the sciences in recent times.” However, few mathematicians of the time were equipped to understand the young scholar’s complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel’s discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

User’s Reviews

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

⭐If Godel proved that no sufficiently complex system, i.e one that is capable of arithmetic, can prove its own consistency or if you assume the system is consistent there will always exist (infinitely many) true statements that cannot be deduced from its axioms, in what system did he prove it in? Is that system consistent? In what sense is the Godel statement true if not by proof? You’ll have hundreds of questions popping in your mind every few minutes, and this short book does a very good job of tackling most of them.Godel numbering is a way to map all the expressions generated by the successive application of axioms back onto numbers, which are themselves instantiated as a “model” of the axioms. The hard part of it is to do this by avoiding the “circular hell”. Russell in Principia Mathematica tried hard to avoid the kind of paradoxes like “Set of all elements which do not belong to the set”. Godel’s proof tries hard to avoid more complicated paradoxes like this :Let p = “Is a sum of two primes” be a property some numbers might possess. This property can be stated precisely using axioms, and symbols can be mapped to numbers. ( for e.g open a text file, write down the statement and look at its ASCII representation ). The let n(p) be the number corresponding to p. If n(p) satisfies p, then we say n(p) is Richardian, else not. Being Richardian itself is a meta-mathematical property r = “A number which satisfies the property described by its reverse ASCII representation”. Note that it is a proper statement represented by the symbols that make up your axioms. Now, you ask if n(r) is Richardian, and the usual problem emerges : n(r) is Richardian iff it is not Richardian. This apparent conundrum, as the authors say, is a hoax. We wanted to represent arithmetical statements as numbers, but switched over to representing meta-mathematical statements as numbers. Godel’s proof avoid cheating like this by carefully mirroring all meta-mathematical statements within the arithmetic, and not just conflating the two. Four parts to it.1. Construct a meta-mathemtical formula G that represents “The formula G is not demonstratable”. ( Like Richardian )2. G is demonstrable if and only if ~G is demonstrable ( Like Richardian)3. Though G is not demonstrable, G is true in the sense that it asserts a certain arithmetical property which can be exactly defined. ( Unlike Richardian ).4. Finally, Godel showed that the meta-mathematical statement “if `Arithmetic is consistent’ then G follows” is demonstrable. Then he showed that “Arithmetic is consistent” is not demonstrable.It took me a while to pour over the details, back and forth between pages. I’m still not at the level where I can explain the proof to anyone clearly, but I intend to get there eventually. Iterating is the key.When I first came across Godel’s theorem, I was horrified, dismayed, disillusioned and above all confounded – how can successively applying axioms over and over not fill up the space of all theorems? Now, I’m slowly recuperating. One non-mathematical, intuitive, consoling thought that keeps popping into my mind is : If the axioms to describe arithmetic ( or something of a higher, but finite complexity ) were consistent and complete, then why those axioms? Who ordained them? Why not something else? If it turned out that way, then the question of which is more fundamental : physics or logic would be resolved. I would be shocked if it were possible to decouple the two and rank them – one as more fundamental than the other. I’m very slowly beginning to understand why Godel’s discovery was a shock to me.You see, I’m good at rolling with it while I’m working away, but deep down, I don’t believe in Mathematical platonism, or logicism, or formalism or any philosophical ideal that tries to universally quantify.Kindle Edition – I would advise against the Kindle edition as reading anything with a decent bit of math content isn’t a very linear process. Turning pages, referring to footnotes and figures isn’t easy on the Kindle.

⭐This short book shows Godel’s proof of the incompleteness of axiomatic systems that may be consistent. It reads easily – until it doesn’t. It’s very, very helpful to feel comfortable with “~p v q” to understand the ingenious proof based on numbers, but the first 45 pages plus introduction are understandable to just about everyone.The text may be too short to give non-fanatics much insight into what the fuss is about anyway. The book’s summary is a mere four pages.Let me add this. Immanuel Kant was important back in the late 1700’s. Really important. He showed how mental math had real meaning in the material world and that logic has remained fixed for millennia. Kant’s statements marked the end of classicism. In the 1800’s, Gauss, Riemann and others showed how math does not AUTOMATICALLY (a priori) describe the real world (synthetic), and Boole developed symbolic logic (extreme a priori), that seemed to subsume math until the Godel showed that this was a sterile end also. In short, math became extremely separated from reality and Godel then detached it from being a “simple” mental construction.The question remains what does math and other mental systems have to do with the real world. People interpret the proof differently. This book only slightly helps resolve that problem. The early twentieth century gave three big ideas that may be very limited or very pervasive in reality. It’s not really clear: * Relativity Theory (Einstein) * Indeterminacy Principle (Heisenberg) * Incompleteness Theorem (Godel)To get into a modern proof, that is nothing like the simple stuff taught in high school, read this book. To get into Godel, read Goldstein.

⭐I redid my review (now July 2006) after your 50:50 votes on helpfulness. I think you needed more content to the review and less ebullience. So here it is… In the interim, I have read other treatments of Gödel’s proof (including the Dover book of Gödel’s article itself also with an introduction, Beyond Numeracy, The Advent of the Algorithm [ref below], and several others). What stands out in THIS book, though, is the extreme thoroughness of explaining to you the context in which Gödel was working at the time. This book is unique in its dedication to getting you to a concrete understanding of — and appreciation for — the background and context. In fact making sure you get the context appreciation takes up about 2/3rd’s of the book! Of course the book is thorough on the Proof itself too. Is that part easy? No, it’s still not. But you won’t be left at all vague on what the proof is like. The only other book that is as good on the CONTEXT of Gödel’s proof is The Advent of the Algorithm. The Advent of the Algorithm is also excellent on how others took, and “ran-with”, Gödel’s results. As for which edition of this book (Gödel’s Proof) to get, the new addition has Hofstaddter’s introduction. That intro adds value for sentimentality (if you should so find his story about his reading the book and his subsequent friendship with Nagel) and Hofstadter’s own ebullience, but the book is virtually identical otherwise with its 1959 edition. It would be perfectly good — you’ll miss nothing — if you bought a cheap 1959 edition. For a good complimentary book, get also The Advent of the Algorithm by David Berlinski (2000) ISBN 0 15 100338 6 or ISBN 0 15 601391 6 (pbk). You can read my review on that book too if you like.

⭐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.

⭐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.

⭐This is quite helpful if you are going to tackle Godel’s key text ‘On Formally Undecidable Propositions of “Principia Mathematica” and Related Systems’

⭐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.

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