recursion theory for metamathematics oxford logic guides

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Recursion Theory For Metamathematics

Author : Raymond M. Smullyan
ISBN : 9780195082326
Genre : Social Science
File Size : 22. 73 MB
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This work is a sequel to the author's G�del's Incompleteness Theorems, though it can be read independently by anyone familiar with G�del's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.

Diagonalization And Self Reference

Author : Raymond M. Smullyan
ISBN : 0198534507
Genre : Mathematics
File Size : 75. 83 MB
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The main purpose of this book is to present a unified treatment of fixed points as they occur in Godel's incompleteness proofs, recursion theory, combinatory logic, semantics and metamathematics. It provides a survey of introductory material and a summary of recent research. The first chapters are of an introductory nature, and consist mainly of exercises with solutions given to most of them. The book should be of interest to researchers and graduate students in mathematical logic.

A Beginner S Further Guide To Mathematical Logic

Author : Raymond Smullyan
ISBN : 9789814733014
Genre :
File Size : 88. 84 MB
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This is the final book written by the late great puzzle master and logician, Dr. Raymond Smullyan. This book is a sequel to my Beginner's Guide to Mathematical Logic. The previous volume deals with elements of propositional and first-order logic, contains a bit on formal systems and recursion, and concludes with chapters on Gödel's famous incompleteness theorem, along with related results. The present volume begins with a bit more on propositional and first-order logic, followed by what I would call a "fein" chapter, which simultaneously generalizes some results from recursion theory, first-order arithmetic systems, and what I dub a "decision machine." Then come five chapters on formal systems, recursion theory and metamathematical applications in a general setting. The concluding five chapters are on the beautiful subject of combinatory logic, which is not only intriguing in its own right, but has important applications to computer science. Argonne National Laboratory is especially involved in these applications, and I am proud to say that its members have found use for some of my results in combinatory logic. This book does not cover such important subjects as set theory, model theory, proof theory, and modern developments in recursion theory, but the reader, after studying this volume, will be amply prepared for the study of these more advanced topics. Request Inspection Copy

Physical A Causality

Author : Karl Svozil
ISBN : 9783319708157
Genre : Science
File Size : 37. 70 MB
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This book is open access under a CC BY 4.0 license. This book addresses the physical phenomenon of events that seem to occur spontaneously and without any known cause. These are to be contrasted with events that happen in a (pre-)determined, predictable, lawful, and causal way. All our knowledge is based on self-reflexive theorizing, as well as on operational means of empirical perception. Some of the questions that arise are the following: are these limitations reflected by our models? Under what circumstances does chance kick in? Is chance in physics merely epistemic? In other words, do we simply not know enough, or use too crude levels of description for our predictions? Or are certain events "truly", that is, irreducibly, random? The book tries to answer some of these questions by introducing intrinsic, embedded observers and provable unknowns; that is, observables and procedures which are certified (relative to the assumptions) to be unknowable or undoable. A (somewhat iconoclastic) review of quantum mechanics is presented which is inspired by quantum logic. Postulated quantum (un-)knowables are reviewed. More exotic unknowns originate in the assumption of classical continua, and in finite automata and generalized urn models, which mimic complementarity and yet maintain value definiteness. Traditional conceptions of free will, miracles and dualistic interfaces are based on gaps in an otherwise deterministic universe.

Godel S Incompleteness Theorems

Author : Raymond M. Smullyan
ISBN : 9780195364378
Genre : Computers
File Size : 27. 44 MB
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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.

Twenty Five Years Of Constructive Type Theory

Author : Giovanni Sambin
ISBN : 9780198501275
Genre : Mathematics
File Size : 62. 30 MB
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Martin-Löf Type Theory is both an important and practical formalization and a focus for a charismatic view of the foundations of mathematics. Per Martin-Löf's work has been of huge significance in the fields of logic and the foundations of mathematics, and has important applications in areas such as computing science and linguistics. This volume celebrates the twenty-fifth anniversary of the birth of the subject, and is an invaluable record both of areas of currentactivity and of the early development of the subject. Also published for the first time is one of Per Martin-Löf's earliest papers.

Reductive Logic And Proof Search

Author : David J. Pym
ISBN : 9780198526339
Genre : Mathematics
File Size : 45. 58 MB
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This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects , J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 .

Introduction To Mathematical Logic Sixth Edition

Author : Elliott Mendelson
ISBN : 9781482237788
Genre : Mathematics
File Size : 72. 36 MB
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The new edition of this classic textbook, Introduction to Mathematical Logic, Sixth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Gödel, Church, Kleene, Rosser, and Turing. The sixth edition incorporates recent work on Gödel’s second incompleteness theorem as well as restoring an appendix on consistency proofs for first-order arithmetic. This appendix last appeared in the first edition. It is offered in the new edition for historical considerations. The text also offers historical perspectives and many new exercises of varying difficulty, which motivate and lead students to an in-depth, practical understanding of the material.

Computability Theory

Author : Rebecca Weber
ISBN : 9780821873922
Genre : Mathematics
File Size : 66. 5 MB
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What can we compute--even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoretically? These questions are at the heart of computability theory. The goal of this book is to give the reader a firm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness. Turing machines and partial recursive functions are explored in detail, and vital tools and concepts including coding, uniformity, and diagonalization are described explicitly. From there the material continues with universal machines, the halting problem, parametrization and the recursion theorem, and thence to computability for sets, enumerability, and Turing reduction and degrees. A few more advanced topics round out the book before the chapter on areas of research. The text is designed to be self-contained, with an entire chapter of preliminary material including relations, recursion, induction, and logical and set notation and operators. That background, along with ample explanation, examples, exercises, and suggestions for further reading, make this book ideal for independent study or courses with few prerequisites.

Revenge Of The Liar

Author : J. C. Beall
ISBN : UOM:39015073917695
Genre : Philosophy
File Size : 57. 64 MB
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Fourteen new essays by some of the world's leading experts, together with an extensive introduction, examine the nature of the Liar paradox and its resistance to any attempt to solve it.

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