Gödel's Proof

Author: Ernest Nagel
Publisher: Psychology Press
ISBN: 041504040X
Category : Gödel's theorem
Languages : en
Pages : 118

Book Description
In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, Godelâ€™s Proofby Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity.

Incompleteness

Author: Rebecca Goldstein
Publisher: W. W. Norton & Company
ISBN: 0393327604
Category : Biography & Autobiography
Languages : en
Pages : 299

Book Description
"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.

An Introduction to Gödel's Theorems

Author: Peter Smith
Publisher: Cambridge University Press
ISBN: 1139465937
Category : Mathematics
Languages : en
Pages : 376

Book Description
In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

Author: Kurt Gödel
Publisher: Courier Corporation
ISBN: 0486158403
Category : Mathematics
Languages : en
Pages : 82

Book Description
First English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.

Godel's Incompleteness Theorems

Author: Raymond M. Smullyan
Publisher: Oxford University Press
ISBN: 0195364376
Category : Mathematics
Languages : en
Pages : 156

Book Description
Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

Gödel's Theorem

Author: Torkel Franzén
Publisher: CRC Press
ISBN: 1439876924
Category : Mathematics
Languages : en
Pages : 184

Book Description
"Among the many expositions of Gödel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén 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 an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature." --- John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt Gödel

Gödel's Theorem in Focus

Author: Stuart Shanker
Publisher: Psychology Press
ISBN: 0415045754
Category : Gödel's theorem
Languages : en
Pages : 272

Book Description
A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.

Metamathematics, Machines and Gödel's Proof

Author: N. Shankar
Publisher: Cambridge University Press
ISBN: 9780521585330
Category : Computers
Languages : en
Pages : 224

Book Description
Describes the use of computer programs to check several proofs in the foundations of mathematics.

Forever Undecided

Author: Raymond M. Smullyan
Publisher: Knopf
ISBN: 0307962466
Category : Mathematics
Languages : en
Pages : 286

Book Description
Forever Undecided is the most challenging yet of Raymond Smullyan’s puzzle collections. It is, at the same time, an introduction—ingenious, instructive, entertaining—to Gödel’s famous theorems. With all the wit and charm that have delighted readers of his previous books, Smullyan transports us once again to that magical island where knights always tell the truth and knaves always lie. Here we meet a new and amazing array of characters, visitors to the island, seeking to determine the natives’ identities. Among them: the census-taker McGregor; a philosophical-logician in search of his flighty bird-wife, Oona; and a regiment of Reasoners (timid ones, normal ones, conceited, modest, and peculiar ones) armed with the rules of propositional logic (if X is true, then so is Y). By following the Reasoners through brain-tingling exercises and adventures—including journeys into the “other possible worlds” of Kripke semantics—even the most illogical of us come to understand Gödel’s two great theorems on incompleteness and undecidability, some of their philosophical and mathematical implications, and why we, like Gödel himself, must remain Forever Undecided!

A World Without Time

Author: Palle Yourgrau
Publisher: Basic Books
ISBN: 078673700X
Category : Science
Languages : en
Pages : 226

Book Description
It is a widely known but little considered fact that Albert Einstein and Kurt Godel were best friends for the last decade and a half of Einstein's life. The two walked home together from Princeton's Institute for Advanced Study every day; they shared ideas about physics, philosophy, politics, and the lost world of German science in which they had grown up. By 1949, Godel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist . Einstein endorsed this result-reluctantly, since it decisively overthrew the classical world-view to which he was committed. But he could find no way to refute it, and in the half-century since then, neither has anyone else. Even more remarkable than this stunning discovery, however, was what happened afterward: nothing. Cosmologists and philosophers alike have proceeded with their work as if Godel's proof never existed -one of the greatest scandals of modern intellectual history. A World Without Time is a sweeping, ambitious book, and yet poignant and intimate. It tells the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue from undeserved obscurity the brilliant work they did together.