Gödel’s Proof has ratings and reviews. WarpDrive said: Highly entertaining and thoroughly compelling, this little gem represents a semi-technic.. . Godel’s Proof Ernest Nagel was John Dewey Professor of Philosophy at Columbia In Kurt Gödel published his fundamental paper, “On Formally. UNIVERSITY OF FLORIDA LIBRARIES ” Godel’s Proof Gddel’s Proof by Ernest Nagel and James R. Newman □ r~ ;□□ ii □Bl J- «SB* New York University.
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Goodreads helps you keep track of books you want to read. Want to Read saving…. Want to Read Currently Reading Read. Refresh and try again. Open Preview See a Problem? Thanks for telling us about the problem. Return to Book Page. The award committee described his work in mathematical logic as “one of the greatest contributions to the sciences in recent times.
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 Prooc. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.
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Lists with This Book. Oct 09, WarpDrive rated it it was amazing. Highly entertaining and thoroughly compelling, this little gem represents navel semi-technical but comprehensive and mathematically accurate elucidation of the famous and so often misused and misunderstood Godel’s meta-mathematical results concerning the limits of provability in formal axiomatic theories.
Being relatively short, this book does not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing in theory of computability, Highly entertaining and thoroughly compelling, this little gem represents a semi-technical but comprehensive and mathematically accurate elucidation nage the famous and so often misused and misunderstood Godel’s meta-mathematical results concerning the limits propf provability in formal axiomatic theories.
Being relatively short, this book poof not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing in theory of computability, particularly in the theory of recursive functions, there is a fundamental theorem stating that there are semi-decidable sets sets which can be effectively generatedthat are not fully decidable.
In fact, this is nothing but the Godel’s first theorem expressed in computational terms. As expressed beautifully by Chaitin, uncomputability is the deeper reason for incompleteness.
The reader can get, for example, a quite good appreciation of the genius of Godel’s approach, who understood that mathematics is a universal medium for prooc embedding of patterns or structure of any sort, so that statements seemingly about numbers alone can in fact encode statements about other universes of discourse, including meta-mathematics itself.
And it is precisely by using this fundamental result that Godel could demonstrate his celebrated theorems.
Godel’s incompleteness theorems are about formal provability in a finitistic sense within a specific class of formal systems, rather than about “provability” in an informal sense, or even about provability in mathematics in general. I want here to digress a little from the specific contents of this book, and I want to take the opportunity to dispel at least a couple of the many misconceptions about Godel’s theorems: Given a formal system such as PA or ZFC, the relationship between the axioms and the theorems of the theory is perfectly mechanical and deterministic, and in theory recursively enumerable by a computer program.
Actually, this is one of the pre-requisites for the applicability of Godel’s incompleteness theorems – secondly, the Godel incompleteness theorems must also be put in their correct overall context: Metamathematical arguments establishing the consistency of formal systems such as ZFC have been devised not just by Gentzen, but also by other researchers. For example, we can prove the consistency of ZFC by assuming that there is an inaccessible cardinal.
But, of course, Godel’s theorems imply that this can’t be turned into a proof inside ZFC itself, because ZFC can’t prove the hypothesis that there is an inaccessible cardinal. In gode, terms, we can’t prove the consistency of any sufficiently powerful given formal system from within such system.
This important result states that any first-order theorem which is nayel in all models of a theory must be logically deducible from that theory, and vice versa for example, in abstract algebra any result which is true for all groups, must be deducible from the group axioms.
This is, of course, not an absolute apriori proof of consistency, as originally dreamt by Hilbert, but it is quite an important consideration that should not be forgotten either. This is actually what happened historically, when more sophisticated theories such as ZFC developed out of the naive set theories initially proposed by set theorists.
The fact that there are number-theoretical truths which can not be formally demonstrated within a single jagel formal system in other words, you can’t put all mathematical truths in one single formal axiomatic systemdoes NOT mean that there are truths which are forever incapable of becoming known, or that some sort of mystic human intuition must replace cogent, rigorous proof.
Mathematics is, informationally speaking, infinitely powerful – it can’t be compressed into porof limited, finite set of axioms from which all the mathematical truths can be derived. Anyway, nagep back to this remarkable book, I think that it is one of the best not-fully-technical available treatments of these seminal theorems: It is not complete and a bit dated in parts, but an excellent treatise nevertheless, fully deserving a 5-star rating. View all 15 comments.
Godel’s Proof | Books – NYU Press | NYU Press
This Statement is False. How can anyone wrap oneself around that ancient paradox? A true statement whose unprovability resulted precisely from its truth! With this self-undermining formula godsl his hammer, Godel knocked down the fortress of Principia Mathematica PMso painstakingly constructed by Russell and Whitehead.
He also showed that his method applied to any system whatsoever that tried to accomplish the goals of Principia Mathematica. In effect, then, Godel destroyed the hopes of those who This Statement is False. However, if a formula and its own negation are both formally demonstrable, then PM is not consistent. Accordingly, if PM is consistent, neither G nor not G can be formally derivable from the axioms.
In short, if PM is consistent, then G is a formally undecidable formula. In other words, we cannot deduce all arithmetical truths from the axioms and rules of PM. We are thus compelled to recognize a fundamental limitation concerning the power of formal axiomatic reasoning.
For anyone inclined to believe that the essence of mathematics or any other science is purely formal axiomatic reasoning, this must come as a shocking revelation. Forty-six preliminary definitions, together with several important preliminary propositions, must be mastered before the main results are reached.
Also, this is the book that inspired a teenaged Douglas R. Hofstadter to write his epic opus. For that alone, it is worth a read. View all 23 comments. May 25, Matt rated it really liked it.
I don’t read much math these days, so when I do read it, it’s a little like climbing a steep wall following a winter of sitting in front of a computer. You discover a lot of soreness in muscles you scarcely remember you ever had in the first place. The long and the short of it is that once upon a time, I sorta understood Godel’s incompleteness theorem, and after this modest reading, I sorta understand it again.
More importantly for me, it was fun to try to connect neurons in my poor fuzzy I don’t read much math these days, so when I do read it, it’s a little like climbing a steep wall following a winter of sitting in front of a computer.
More importantly for me, it was fun to try to connect neurons in my poor fuzzy brain, and for a math aficionado, entering a world where it’s assumed that conclusions are merely the logical consequences of initial assumptions and nothing more is a bit like diving into mom’s meatloaf — familiar and comforting.
Godel’s theorem itself is supposedly the big revelation that upset the 19th century mathematics apple cart. As the authors explain, his proof showed that “no final systematization of many important areas of mathematics is attainable,” a potentially unsettling proposition.
But it may be a somewhat comforting assertion. Who would want to have it all figured out anyway? And gravity still works, and the sun still comes up every day, so I think we’ll all get by. By the same reasoning, if you try to simplify a very complex proof, leaving out a lot of the messy details while trying to explain the rest in plain English, you’re necessarily going to miss a lot and there will be important details that are missing.
In this book, the concepts are simplified and the details are left out.
On the plus side, it’s pretty readable for a math book, and the main conclusions are pretty easy to follow. On the minus side, there will necessarily be points to question. They do have a lot of footnotes, which offers some middle ground. Feb 20, Szplug rated it really liked it. My thanks to AC for convincing me to take the plunge and purchase this little gem: It was little touches like the chess analogy for describing the relationship between mathematics and metamathematics, the placing of the Richard paradox in terms that were more pellucid than the valiant effort attempted by Rebecca Goldsteinand the tricky, but effective, explanation of how G My thanks to AC for convincing me to take the plunge and purchase this little gem: What’s more, the appendix led me to dive into the slippery spaghetti of a more in-depth examination of the principles of logical inference and tautologies in absolute proofs.
A wonderful book, one which I am surprisingly happy to have read. Perhaps now I’m ready for DFW’s tour of infinity! I believe it went some ways towards allowing me a clearer understanding of what Nagel and Newman were saying – though they did a magnificent job all on their own of making the entire affair intelligible to this math-rusty reader.
Ce livre comporte trois ouvrages distincts. En secon Ce livre comporte trois ouvrages distincts. Comparaison n’est pas raison.
Je n’y vois pas plus clair. View all 19 comments. In other words, given any consistent set of arithmetical axioms, there are true arithmetical statements that cannot be derived from the set. Godel showed that it is impossible to give a meta-mathematical proof of the consistency of a system comprehensive enough to contain the whole of arithmetic—unless the The Book is the best to explain Godel’s Proof of the Incompleteness Theorem.
Godel showed that it is impossible to give a meta-mathematical proof of the consistency of a system comprehensive enough to contain the whole of arithmetic—unless the proof itself employs rules of inference in certain essential respects different from the Transformation Rules used in deriving theorems within the system. Such a proof may, to be sure, possess great value and importance. However, if the reasoning in it is based on rules of inference much more powerful than the goddel of the arithmetical calculus, so that the consistency of the assumptions in the reasoning fodel as subject to doubt as is the consistency of arithmetic, the proof would yield only a specious victory: The statement does not express an arithmetical fact and does not belong to the formal language of arithmetic; it belongs to meta-mathematics, because it characterizes a certain string of arithmetical signs as being a formula.
According to a standard convention we construct a name for a linguistic expression by placing single quotation marks around it.
Our text adheres to this convention. It is correct to write: Chicago is a populous city.
But it is incorrect to write: To express what is intended by this latter sentence, one must write: Likewise, it is incorrect to write: We must, instead, formulate our intent by: Godel devised a method of representation such that neither the arithmetical formula corresponding to a certain true meta-mathematical statement about the formula, nor the arithmetical formula corresponding to the denial of the statement, is demonstrable within the calculus.