A History of the Theory of Recursive Functions and Computability with Special Reference to the Developments Initiated by Godels Incompleteness Theorems PDF Download

Author: R. G. Adams
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Author: Rod Adams
Publisher: Docent Press
ISBN: 0983700400
Category : Mathematics
Languages : en
Pages : 312

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Book Description
Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

Author: Roman Murawski
Publisher: Springer Science & Business Media
ISBN: 9401728666
Category : Philosophy
Languages : en
Pages : 416

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Book Description
Recursive Functions and Metamathematics deals with problems of the completeness and decidability of theories, using as its main tool the theory of recursive functions. This theory is first introduced and discussed. Then Gödel's incompleteness theorems are presented, together with generalizations, strengthenings, and the decidability theory. The book also considers the historical and philosophical context of these issues and their philosophical and methodological consequences. Recent results and trends have been included, such as undecidable sentences of mathematical content, reverse mathematics. All the main results are presented in detail. The book is self-contained and presupposes only some knowledge of elementary mathematical logic. There is an extensive bibliography. Readership: Scholars and advanced students of logic, mathematics, philosophy of science.

Author: Hartley Rogers
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 526

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

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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.

Author: British Library. Lending Division
Publisher:
ISBN:
Category : Great Britain
Languages : en
Pages : 542

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Issue for Mar. 1981 contains index for Jan.-Mar. 1981 in microfiche form.

Author: Hans Hermes
Publisher: Springer
ISBN: 3662116863
Category : Mathematics
Languages : en
Pages : 255

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The task of developing algorithms to solve problems has always been considered by mathematicians to be an especially interesting and im portant one. Normally an algorithm is applicable only to a narrowly limited group of problems. Such is for instance the Euclidean algorithm, which determines the greatest common divisor of two numbers, or the well-known procedure which is used to obtain the square root of a natural number in decimal notation. The more important these special algorithms are, all the more desirable it seems to have algorithms of a greater range of applicability at one's disposal. Throughout the centuries, attempts to provide algorithms applicable as widely as possible were rather unsuc cessful. It was only in the second half of the last century that the first appreciable advance took place. Namely, an important group of the inferences of the logic of predicates was given in the form of a calculus. (Here the Boolean algebra played an essential pioneer role. ) One could now perhaps have conjectured that all mathematical problems are solvable by algorithms. However, well-known, yet unsolved problems (problems like the word problem of group theory or Hilbert's tenth problem, which considers the question of solvability of Diophantine equations) were warnings to be careful. Nevertheless, the impulse had been given to search for the essence of algorithms. Leibniz already had inquired into this problem, but without success.

Author: Heinz-Dieter Ebbinghaus
Publisher: Springer
ISBN: 3540395962
Category : Mathematics
Languages : en
Pages : 427

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Author: Rolf Herken
Publisher: Oxford University Press, USA
ISBN:
Category : Computers
Languages : en
Pages : 708

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This volume commemorates the work of Alan Turing, who not only introduced the most influential concept of a machine model of effective computability, but who also anticipated in his work the diversity of topics brought together here. Among his major contributions, Turing's "On Computable Numbers, With an Application to the Entscheidungsproblem," first published in 1937, is acknowledged as a landmark of the computer age. Part I of this volume explores historical aspects with essays on background, on Turing's work, and on subsequent developments. Part II contains an extensive series of essays on the influence and applications of these ideas in mathematics, mathematical logic, philosophy of mathematics, computer science, artificial intelligence, philosophy of language, philosophy of mind, and physics.

Author: Richard Zach
Publisher: Createspace Independent Publishing Platform
ISBN: 9781548138080
Category :
Languages : en
Pages : 228

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Book Description
A textbook on recursive function theory and G�del's incompleteness theorems. Also covers models of arithmetic and second-order logic.