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Author : Peter Smith
Publisher : Cambridge University Press
Page : 376 pages
File Size : 44,5 Mb
Release : 2007-07-26
Category : Mathematics
ISBN : 9780521857840
Peter Smith examines Gödel's Theorems, how they were established and why they matter.
Author : Peter Smith
Publisher : Unknown
Page : 388 pages
File Size : 51,9 Mb
Release : 2020
Category : Electronic
ISBN : 9798673862131
Author : Lorenz Halbeisen,Regula Krapf
Publisher : Springer Nature
Page : 236 pages
File Size : 49,5 Mb
Release : 2020-10-16
Category : Mathematics
ISBN : 9783030522797
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.
Author : Torkel Franzén
Publisher : CRC Press
Page : 182 pages
File Size : 43,8 Mb
Release : 2005-06-06
Category : Mathematics
ISBN : 9781439876923
"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
Author : Richard Zach
Publisher : Createspace Independent Publishing Platform
Page : 228 pages
File Size : 46,6 Mb
Release : 2017-06-15
Category : Electronic
ISBN : 1548138088
A textbook on recursive function theory and G�del's incompleteness theorems. Also covers models of arithmetic and second-order logic.
Author : Ernest Nagel,James R. Newman
Publisher : Routledge
Page : 109 pages
File Size : 44,7 Mb
Release : 2012-11-12
Category : Philosophy
ISBN : 9781134953998
The first book to present a readable explanation of Godel's theorem to both scholars and non-specialists, this is a gripping combination of science and accessibility, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity.
Author : Kurt Gödel
Publisher : Courier Corporation
Page : 82 pages
File Size : 52,9 Mb
Release : 2012-05-24
Category : Mathematics
ISBN : 9780486158402
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.
Author : Paolo Mancosu,Sergio Galvan,Richard Zach
Publisher : Oxford University Press
Page : 431 pages
File Size : 43,9 Mb
Release : 2021
Category : Philosophy
ISBN : 9780192895936
An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.
Author : S.G. Shanker
Publisher : Taylor & Francis
Page : 271 pages
File Size : 49,8 Mb
Release : 2012-08-21
Category : Philosophy
ISBN : 9781134947980
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.
Author : Rebecca Goldstein
Publisher : W. W. Norton & Company
Page : 299 pages
File Size : 42,6 Mb
Release : 2006-01-31
Category : Biography & Autobiography
ISBN : 9780393327601
"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
Author : Peter Smith
Publisher : Unknown
Page : 406 pages
File Size : 49,6 Mb
Release : 2014-05-14
Category : Logic, Symbolic and mathematical
ISBN : 1107336589
A clear and accessible treatment of Godel's famous, intriguing, but much misunderstood incompleteness theorems, extensively revised in a second edition."
Author : Harry J. Gensler
Publisher : Unknown
Page : 94 pages
File Size : 50,6 Mb
Release : 1984
Category : Mathematics
ISBN : UOM:39015049391991
This helpful volume explains and proves Godel's theorem, which states that arithmetic cannot be reduced to any axiomatic system. Written simply and directly, this book is intended for the student and general reader and presumes no specialized knowledge of mathematics or logic.
Author : Raymond M. Smullyan
Publisher : Oxford University Press
Page : 156 pages
File Size : 41,8 Mb
Release : 1992-08-20
Category : Mathematics
ISBN : 9780195364378
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.
Author : Richard E. Hodel
Publisher : Courier Corporation
Page : 514 pages
File Size : 55,8 Mb
Release : 2013-01-01
Category : Mathematics
ISBN : 9780486497853
This comprehensive overview ofmathematical logic is designedprimarily for advanced undergraduatesand graduate studentsof mathematics. The treatmentalso contains much of interest toadvanced students in computerscience and philosophy. Topics include propositional logic;first-order languages and logic; incompleteness, undecidability,and indefinability; recursive functions; computability;and Hilbert’s Tenth Problem.Reprint of the PWS Publishing Company, Boston, 1995edition.
Author : Christopher C. Leary,Lars Kristiansen
Publisher : Lulu.com
Page : 382 pages
File Size : 49,5 Mb
Release : 2015
Category : Education
ISBN : 9781942341079
At the intersection of mathematics, computer science, and philosophy, mathematical logic examines the power and limitations of formal mathematical thinking. In this expansion of Leary's user-friendly 1st edition, readers with no previous study in the field are introduced to the basics of model theory, proof theory, and computability theory. The text is designed to be used either in an upper division undergraduate classroom, or for self study. Updating the 1st Edition's treatment of languages, structures, and deductions, leading to rigorous proofs of Gödel's First and Second Incompleteness Theorems, the expanded 2nd Edition includes a new introduction to incompleteness through computability as well as solutions to selected exercises.