Search Results for "goedel-s-theorem"

Finite sets and Gödel's incompleteness theorems

Finite sets and Gödel's incompleteness theorems

  • Author: S. Świerczkowski
  • Publisher: N.A
  • ISBN: N.A
  • Category: Gödel's theorem
  • Page: 58
  • View: 1851
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An Introduction to Gödel's Theorems

An Introduction to Gödel's Theorems

  • Author: Peter Smith
  • Publisher: Cambridge University Press
  • ISBN: 9781139465939
  • Category: Mathematics
  • Page: N.A
  • View: 874
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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.

Kurt Gödel: Collected Works: Volume I

Kurt Gödel: Collected Works: Volume I

Publications 1929-1936

  • Author: Kurt Gödel,Milton Jonathan Slocum,S. Feferman
  • Publisher: Oxford University Press
  • ISBN: 9780195039641
  • Category: Mathematics
  • Page: 490
  • View: 8007
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Kurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass. These long-awaited final two volumes contain Gödel's correspondence of logical, philosophical, and scientific interest. Volume IV covers A to G, with H to Z in volume V; in addition, Volume V contains a full inventory of Gödel's Nachlass. L All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited. Kurt Gödel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Gödel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century.

Godel's Theorem in Focus

Godel's Theorem in Focus

  • Author: S.G. Shanker
  • Publisher: Routledge
  • ISBN: 1134947976
  • Category: Philosophy
  • Page: 272
  • View: 2033
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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.

Computability and Logic

Computability and Logic

  • Author: George S. Boolos,John P. Burgess,Richard C. Jeffrey
  • Publisher: Cambridge University Press
  • ISBN: 9780521007580
  • Category: Philosophy
  • Page: 356
  • View: 2106
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This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's Incompleteness Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer.

The Formal Semantics of Programming Languages

The Formal Semantics of Programming Languages

An Introduction

  • Author: Glynn Winskel
  • Publisher: MIT Press
  • ISBN: 9780262731034
  • Category: Computers
  • Page: 361
  • View: 5687
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The Formal Semantics of Programming Languages provides the basic mathematical techniques necessary for those who are beginning a study of the semantics and logics of programming languages. These techniques will allow students to invent, formalize, and justify rules with which to reason about a variety of programming languages. Although the treatment is elementary, several of the topics covered are drawn from recent research, including the vital area of concurency. The book contains many exercises ranging from simple to miniprojects.Starting with basic set theory, structural operational semantics is introduced as a way to define the meaning of programming languages along with associated proof techniques. Denotational and axiomatic semantics are illustrated on a simple language of while-programs, and fall proofs are given of the equivalence of the operational and denotational semantics and soundness and relative completeness of the axiomatic semantics. A proof of Godel's incompleteness theorem, which emphasizes the impossibility of achieving a fully complete axiomatic semantics, is included. It is supported by an appendix providing an introduction to the theory of computability based on while-programs. Following a presentation of domain theory, the semantics and methods of proof for several functional languages are treated. The simplest language is that of recursion equations with both call-by-value and call-by-name evaluation. This work is extended to lan guages with higher and recursive types, including a treatment of the eager and lazy lambda-calculi. Throughout, the relationship between denotational and operational semantics is stressed, and the proofs of the correspondence between the operation and denotational semantics are provided. The treatment of recursive types - one of the more advanced parts of the book - relies on the use of information systems to represent domains. The book concludes with a chapter on parallel programming languages, accompanied by a discussion of methods for specifying and verifying nondeterministic and parallel programs.

Proceedings of the National Academy of Sciences, India

Proceedings of the National Academy of Sciences, India

Section A.

  • Author: National Academy of Sciences, India
  • Publisher: N.A
  • ISBN: N.A
  • Category: Physical sciences
  • Page: N.A
  • View: 4391
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A Sample Survey

A Sample Survey

Cultural Resources on Potential Agricultural Development Lands of the Gila River Indian Reservation

  • Author: William S. Marmaduke,Laverne Conway
  • Publisher: N.A
  • ISBN: N.A
  • Category: Archaeological surveying
  • Page: 467
  • View: 1077
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Directions in Chaos — Volume 1

Directions in Chaos — Volume 1

  • Author: B-L Hao
  • Publisher: World Scientific
  • ISBN: 9814513717
  • Category:
  • Page: 368
  • View: 7000
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This volume, the first of a two-volume book, consists of a collection of comprehensive reviews and lectures written by active researchers on topics in chaotic phenomena. Contents:Directions in Classical Chaos (J Ford)Nonlinear Resonance and Chaos in Conservative Systems (L Reichl & W-M Zheng)Generalized Renormalization Group Analysis of Period-Doubling Bifurcations (K L Liu & K Young)Application of Dimension Algorithms to Experimental Chaos (G Mayer-Kress)Emergence of Chaos in Laser Systems and the Development of Diagnostic Techniques (J Tredicce & L Narducci)Dissipative Classical and Quantum Dynamics: the Morse Oscillator (J M Yuan)Transitions to Chaos in Higher Dimensions (B Hu & J-M Mao)Phenomenology of Spatio-Temporal Chaos (J Crutchfield & K Kaneko) Readership: Condensed matter physicists, applied mathematicians and computer scientists. Keywords:Classical Chaos;Nonlinear Resonance;Quantum Dynamics;Spatio-Temporal ChaosReview:“This is an outstanding book which can be recommended to anyone interested in chaos.”Professor W J Firth Contemporary Physics, 1989

Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets

Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets

  • Author: David Papineau
  • Publisher: OUP Oxford
  • ISBN: 0191656240
  • Category: Philosophy
  • Page: 224
  • View: 1999
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This book is designed to explain the technical ideas that are taken for granted in much contemporary philosophical writing. Notions like denumerability, modal scope distinction, Bayesian conditionalization, and logical completeness are usually only elucidated deep within difficult specialist texts. By offering simple explanations that by-pass much irrelevant and boring detail, Philosophical Devices is able to cover a wealth of material that isnormally only available to specialists. The book contains four sections, each of three chapters. The first section is about sets and numbers, starting with the membership relation and ending with the generalized continuum hypothesis. The second is about analyticity, a prioricity, and necessity. The third is about probability, outlining the difference between objective and subjective probability and exploring aspects of conditionalization and correlation. The fourth deals with metalogic, focusing on the contrast between syntax andsemantics, and finishing with a sketch of Gödels theorem. Philosophical Devices will be useful for university students who have got past the foothills of philosophy and are starting to read more widely, but it does not assume any prior expertise. All the issues discussed are intrinsically interesting, and often downright fascinating. It can be read with pleasure and profit by anybody who is curious about the technical infrastructure of contemporary philosophy.