Recursive Functionals
Recursive Functionals

Author: L.E. Sanchis

Publisher: Elsevier

Published: 1992-05-18

Total Pages: 276

ISBN-13: 9780080887173

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This work is a self-contained elementary exposition of the theory of recursive functionals, that also includes a number of advanced results. Although aiming basically at a theory of higher order computability, attention is restricted to second order functionals, where the arguments are numerical functions and the values, when defined, are natural numbers. This theory is somewhat special, for to some extent it can be reduced to first order theory, but when properly extended and relativized it requires the full machinery of higher order computations. In the theory of recursive monotonic functionals the author formulates a reasonable notion of computation which provides the right frame for what appears to be a convincing form of the extended Church's thesis. At the same time, the theory provides sufficient room to formulate the classical results that are usually derived in terms of singular functionals. Presented are complete proofs of Gandy's selector theorem, Kleene's theorem on hyperarithmetical predicates, and Grilliot's theorem on effectively discontinuous functionals.

Formalized Recursive Functionals and Formalized Realizability
Formalized Recursive Functionals and Formalized Realizability

Author: Stephen Cole Kleene

Publisher: American Mathematical Soc.

Published: 1969

Total Pages: 110

ISBN-13: 0821812890

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This monograph carries out the program which the author formulated in earlier work, the formalization of the theory of recursive functions of type 0 and 1 and of the theory of realizability.

Recursive Functions and Metamathematics
Recursive Functions and Metamathematics

Author: Roman Murawski

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 416

ISBN-13: 9401728666

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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.

Computability
Computability

Author: Nigel Cutland

Publisher: Cambridge University Press

Published: 1980-06-19

Total Pages: 268

ISBN-13: 9780521294652

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What can computers do in principle? What are their inherent theoretical limitations? The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function - a function whose values can be calculated in an automatic way.

Super-Recursive Algorithms
Super-Recursive Algorithms

Author: Mark Burgin

Publisher: Springer Science & Business Media

Published: 2006-12-21

Total Pages: 314

ISBN-13: 0387268065

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* The first exposition on super-recursive algorithms, systematizing all main classes and providing an accessible, focused examination of the theory and its ramifications * Demonstrates how these algorithms are more appropriate as mathematical models for modern computers and how they present a better framework for computing methods * Develops a new practically-oriented perspective on the theory of algorithms, computation, and automata, as a whole

An Introduction to Gödel's Theorems
An Introduction to Gödel's Theorems

Author: Peter Smith

Publisher: Cambridge University Press

Published: 2007-07-26

Total Pages: 376

ISBN-13: 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.