Godel's Incompleteness Theorems

Godel's Incompleteness Theorems
Title Godel's Incompleteness Theorems PDF eBook
Author Raymond M. Smullyan
Publisher Oxford University Press
Pages 156
Release 1992-08-20
Genre Mathematics
ISBN 0195364376

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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.

An Introduction to Gödel's Theorems

An Introduction to Gödel's Theorems
Title An Introduction to Gödel's Theorems PDF eBook
Author Peter Smith
Publisher Cambridge University Press
Pages 376
Release 2007-07-26
Genre Mathematics
ISBN 1139465937

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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.

Incompleteness

Incompleteness
Title Incompleteness PDF eBook
Author Rebecca Goldstein
Publisher W. W. Norton & Company
Pages 299
Release 2006-01-31
Genre Biography & Autobiography
ISBN 0393327604

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

Gödel's Theorem in Focus

Gödel's Theorem in Focus
Title Gödel's Theorem in Focus PDF eBook
Author Stuart Shanker
Publisher Psychology Press
Pages 272
Release 1989
Genre Gödel's theorem
ISBN 0415045754

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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.

Gödel's Proof

Gödel's Proof
Title Gödel's Proof PDF eBook
Author Ernest Nagel
Publisher Psychology Press
Pages 118
Release 1989
Genre Gödel's theorem
ISBN 041504040X

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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, Godel’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.

Gödel's Theorem

Gödel's Theorem
Title Gödel's Theorem PDF eBook
Author Torkel Franzén
Publisher CRC Press
Pages 182
Release 2005-06-06
Genre Mathematics
ISBN 1439876924

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"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 Theorems and Zermelo's Axioms

Gödel's Theorems and Zermelo's Axioms
Title Gödel's Theorems and Zermelo's Axioms PDF eBook
Author Lorenz Halbeisen
Publisher Springer Nature
Pages 236
Release 2020-10-16
Genre Mathematics
ISBN 3030522792

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This book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Gödel’s classical completeness and incompleteness theorems. In particular, the book includes a full proof of Gödel’s second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on the Zermelo’s axioms, containing a presentation of Gödel’s constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers. The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory. Each chapter concludes with a list of exercises.