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| kleene metamathematics: Introduction to Metamathematics S.C. Kleene, 1980-01-01 Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Gadel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic, at least a turning point after which nothing was ever the same. Kleene was an important figure in logic, and lived a long full life of scholarship and teaching. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gade1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Gadel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education. |
| kleene metamathematics: Introduction to Metamathematics Stephen Cole Kleene, 1964 |
| kleene metamathematics: Mathematical Logic Stephen Cole Kleene, 2013-04-22 Contents include an elementary but thorough overview of mathematical logic of 1st order; formal number theory; surveys of the work by Church, Turing, and others, including Gödel's completeness theorem, Gentzen's theorem, more. |
| kleene metamathematics: Sets, Models and Proofs Ieke Moerdijk, Jaap van Oosten, 2018-11-23 This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year. |
| kleene metamathematics: Introduction to Elementary Mathematical Logic Abram Aronovich Stolyar, 1984-01-01 This lucid, non-intimidating presentation by a Russian scholar explores propositional logic, propositional calculus, and predicate logic. Topics include computer science and systems analysis, linguistics, and problems in the foundations of mathematics. Accessible to high school students, it also constitutes a valuable review of fundamentals for professionals. 1970 edition. |
| kleene metamathematics: Introduction to Logic Alfred Tarski, 2013-07-04 This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout. |
| kleene metamathematics: Principia Mathematica Alfred North Whitehead, Bertrand Russell, 1927 The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century. |
| kleene metamathematics: Propositional and Predicate Calculus: A Model of Argument Derek Goldrei, 2005-12-27 Designed specifically for guided independent study. Features a wealth of worked examples and exercises, many with full teaching solutions, that encourage active participation in the development of the material. It focuses on core material and provides a solid foundation for further study. |
| kleene metamathematics: The Undecidable Martin Davis, 2004-01-01 A valuable collection both for original source material as well as historical formulations of current problems. — The Review of Metaphysics Much more than a mere collection of papers. A valuable addition to the literature. — Mathematics of Computation An anthology of fundamental papers on undecidability and unsolvability by major figures in the field , this classic reference is ideally suited as a text for graduate and undergraduate courses in logic, philosophy, and foundations of mathematics. It is also appropriate for self-study. The text opens with Godel's landmark 1931 paper demonstrating that systems of logic cannot admit proofs of all true assertions of arithmetic. Subsequent papers by Godel, Church, Turing, and Post single out the class of recursive functions as computable by finite algorithms. Additional papers by Church, Turing, and Post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by Kleene and Post includes initiation of the classification theory of unsolvable problems. Supplementary items include corrections, emendations, and added commentaries by Godel, Church, and Kleene for this volume's original publication, along with a helpful commentary by the editor. |
| kleene metamathematics: Forever Undecided Raymond M. Smullyan, 2012-07-04 Forever Undecided is the most challenging yet of Raymond Smullyan’s puzzle collections. It is, at the same time, an introduction—ingenious, instructive, entertaining—to Gödel’s famous theorems. With all the wit and charm that have delighted readers of his previous books, Smullyan transports us once again to that magical island where knights always tell the truth and knaves always lie. Here we meet a new and amazing array of characters, visitors to the island, seeking to determine the natives’ identities. Among them: the census-taker McGregor; a philosophical-logician in search of his flighty bird-wife, Oona; and a regiment of Reasoners (timid ones, normal ones, conceited, modest, and peculiar ones) armed with the rules of propositional logic (if X is true, then so is Y). By following the Reasoners through brain-tingling exercises and adventures—including journeys into the “other possible worlds” of Kripke semantics—even the most illogical of us come to understand Gödel’s two great theorems on incompleteness and undecidability, some of their philosophical and mathematical implications, and why we, like Gödel himself, must remain Forever Undecided! |
| kleene metamathematics: The Cambridge Quintet John L. Casti, 1999-04-23 In this narrative tour de force, gifted scientist and author John L. Casti contemplates an imaginary evening of intellectual inquiry—a sort of “My Dinner with” not Andre, but five of the most brilliant thinkers of the twentieth century.Imagine, if you will, one stormy summer evening in 1949, as novelist and scientist C. P. Snow, Britain's distinguished wartime science advisor and author of The Two Cultures, invites four singular guests to a sumptuous seven-course dinner at his alma mater, Christ's College, Cambridge, to discuss one of the emerging scientific issues of the day: Can we build a machine that could duplicate human cognitive processes? The distinguished guest list for Snow's dinner consists of physicist Erwin Schrodinger, inventor of wave mechanics; Ludwig Wittgenstein, the famous twentieth-century philosopher of language, who posited two completely contradictory theories of human thought in his lifetime; population geneticist/science popularizer J.B.S. Haldane; and Alan Turing, the mathematician/codebreaker who formulated the computing scheme that foreshadowed the logical structure of all modern computers. Capturing not only their unique personalities but also their particular stands on this fascinating issue, Casti dramatically shows what each of these great men might have argued about artificial intelligence, had they actually gathered for dinner that midsummer evening.With Snow acting as referee, a lively intellectual debate unfolds. Philosopher Wittgenstein argues that in order to become conscious, a machine would have to have life experiences similar to those of human beings—such as pain, joy, grief, or pleasure. Biologist Haldane offers the idea that mind is a separate entity from matter, so that regardless of how sophisticated the machine, only flesh can bond with that mysterious force called intelligence. Both physicist Schrodinger and, of course, computer pioneer Turing maintain that it is not the substance, but rather the organization of that substance, that makes a mind conscious.With great verve and skill, Casti recreates a unique and thrilling moment of time in the grand history of scientific ideas. Even readers who have already formed an opinion on artificial intelligence will be forced to reopen their minds on the subject upon reading this absorbing narrative. After almost four decades, the solutions to the epic scientific and philosophical problems posed over this meal in C. P. Snow's old rooms at Christ's College remains tantalizingly just out of reach, making this adventure into scientific speculation as valid today as it was in 1949. |
| kleene metamathematics: The Logic of Provability George Boolos, 1995-04-28 Boolos, a pre-eminent philosopher of mathematics, investigates the relationship between provability and modal logic. |
| kleene metamathematics: Understanding Syntax Maggie Tallerman, 2014-11-13 Assuming no prior knowledge, Understanding Syntax illustrates the major concepts, categories and terminology associated with the study of cross-linguistic syntax. A theory-neutral and descriptive viewpoint is taken throughout. Starting with an overview of what syntax is, the book moves on to an explanation of word classes (such as noun, verb, adjective) and then to a discussion of sentence structure in the world’s languages. Grammatical constructions and relationships between words in a clause are explained and thoroughly illustrated, including grammatical relations such as subject and object; function-changing processes such as the passive and antipassive; case and agreement processes, including both ergative and accusative alignments; verb serialization; head-marking and dependent-marking grammars; configurational and non-configurational languages; questions and relative clauses. The final chapter explains and illustrates the principles involved in writing a brief syntactic sketch of a language, enabling the reader to construct a grammatical sketch of a language known to them. Data from approximately 100 languages appears in the text, with languages representing widely differing geographical areas and distinct language families. The book will be essential for courses in cross-linguistic syntax, language typology, and linguistic fieldwork, as well as for basic syntactic description. |
| kleene metamathematics: Logic and Scientific Methods Maria Luisa Dalla Chiara, Kees Doets, Daniele Mundici, Johan van Benthem, 1996-12-31 This is the first of two volumes comprising the papers submitted for publication by the invited participants to the Tenth International Congress of Logic, Methodology and Philosophy of Science, held in Florence, August 1995. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science. The invited lectures published in the two volumes demonstrate much of what goes on in the fields of the Congress and give the state of the art of current research. The two volumes cover the traditional subdisciplines of mathematical logic and philosophical logic, as well as their interfaces with computer science, linguistics and philosophy. Philosophy of science is broadly represented, too, including general issues of natural sciences, social sciences and humanities. The papers in Volume One are concerned with logic, mathematical logic, the philosophy of logic and mathematics, and computer science. |
| kleene metamathematics: Introduction to Metamathematics, by Stephen Cole Kleene,... Stephen Cole Kleene, 1952 |
| kleene metamathematics: A Friendly Introduction to Mathematical Logic Christopher C. Leary, Lars Kristiansen, 2015 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. |
| kleene metamathematics: Logic and Structure Dirk van Dalen, 2013-11-11 Logic appears in a 'sacred' and in a 'profane' form. The sacred form is dominant in proof theory, the profane form in model theory. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. set theory and recursion theory. For one reason or another, such as the discovery of the set theoretical paradoxes (Cantor, Russell), or the definability paradoxes (Richard, Berry), a subject is treated for some time with the utmost awe and diffidence. As a rule, however, sooner or later people start to treat the matter in a more free and easy way. Being raised in the 'sacred' tradition, I was greatly surprised (and some what shocked) when I observed Hartley Rogers teaching recursion theory to mathema ticians as if it were just an ordinary course in, say, linear algebra or algebraic topology. In the course of time I have come to accept his viewpoint as the didac tically sound one: before going into esoteric niceties one should develop a certain feeling for the subject and obtain a reasonable amount of plain working knowledge. For this reason I have adopted the profane attitude in this introductory text, reserving the more sacred approach for advanced courses. Readers who want to know more about the latter aspect of logic are referred to the immortal texts of Hilbert-Bernays or Kleene. |
| kleene metamathematics: Formal Syntax and Semantics of Programming Languages Kenneth Slonneger, Barry L. Kurtz, 1995 With this book, readers with a basic grounding in discreet mathematics will be able to understand the practical applications of these difficult concepts. The book presents the typically difficult subject of formal methods in an informal, easy-to-follow manner. A laboratory component is integrated throughout the text. |
| kleene metamathematics: Handbook of Philosophical Logic Dov M. Gabbay, Franz Guenthner, 2013-04-17 It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good. ! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise, The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other. |
| kleene metamathematics: The New Yearbook for Phenomenology and Phenomenological Philosophy Burt Hopkins, Steven Crowell, 2015-03-19 The New Yearbook for Phenomenology and Phenomenological Philosophy provides an annual international forum for phenomenological research in the spirit of Husserl's groundbreaking work and the extension of this work by such figures as Scheler, Heidegger, Sartre, Levinas, Merleau-Ponty and Gadamer. |
| kleene metamathematics: Experience and Theory Stephan Korner, 2013-04-15 Originally published in 1966. This volume analyzes the general structure of scientific theories, their relation to experience and to non-scientific thought. Part One is concerned with the logic underlying empirical discourse before its subjection to the various constraints, imposed by the logico-mathematical framework of scientific theories upon their content. Part Two is devoted to an examination of this framework and, in particular, to showing that the deductive organization of a field of experience is by that very act a modification of empirical discourse and an idealization of its subject matter. Part Three analyzes the concordance between theories and experience and the relevance of science to moral and religious beliefs. |
| kleene metamathematics: Higher-Order Computability John Longley, Dag Normann, 2015-11-06 This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers |
| kleene metamathematics: Automata Theory University of Michigan. Engineering Summer Conferences, 1963 |
| kleene metamathematics: Perspectives on the History of Mathematical Logic Thomas Drucker, 2009-05-21 This volume offers insights into the development of mathematical logic over the last century. Arising from a special session of the history of logic at an American Mathematical Society meeting, the chapters explore technical innovations, the philosophical consequences of work during the period, and the historical and social context in which the logicians worked. The discussions herein will appeal to mathematical logicians and historians of mathematics, as well as philosophers and historians of science. “...this is an important book. It exposes the richness of ideas and viewpoints, the difficult and not always direct pathways taken in the development of mathematical logic in the last century, and the various factors which did and continue to affect that development.” Modern Logic |
| kleene metamathematics: Handbook of Philosophical Logic D.M. Gabbay, Franz Guenthner, 2006-01-17 The ninth volume of the Second Edition contains major contributions on Rewriting Logic as a Logical and Semantic Framework, Logical Frameworks, Proof Theory and Meaning, Goal Directed Deductions, Negations, Completeness and Consistency as well as Logic as General Rationality. Audience: Students and researchers whose work or interests involve philosophical logic and its applications. |
| kleene metamathematics: Computer Science Logic Erich Grädel, Reinhard Kahle, 2009-08-28 This book constitutes the proceedings of the 23rd International Workshop on Computer Science Logic, CSL 2009, held in Coimbra, Portugal, in September 2009. The 34 papers presented together with 5 invited talks were carefully reviewed and selected from 89 full paper submissions. All current aspects of logic in computer science are addressed, ranging from foundational and methodological issues to application issues of practical relevance. The book concludes with a presentation of this year's Ackermann award, the EACSL Outstanding Dissertation Award for Logic in Computer Science. |
| kleene metamathematics: Applications of Discrete and Continuous Fourier Analysis H. Joseph Weaver, 1992 |
| kleene metamathematics: Advances in Mathematics: Theory, Methods & Applications Akshay Kumar, Mangey Ram, 2025-06-10 This book is an excellent collection of various topics of mathematics which include numerical methods, integral equations, and differential equations. The book is recommended to readers to refresh their understanding of applied mathematics with theory and applications. It will be useful to students, researchers, and practitioners working in applied and computational mathematics. |
| kleene metamathematics: Church's Thesis After 70 Years Adam Olszewski, Jan Wolenski, Robert Janusz, 2013-05-02 Church's Thesis (CT) was first published by Alonzo Church in 1935. CT is a proposition that identifies two notions: an intuitive notion of a effectively computable function defined in natural numbers with the notion of a recursive function. Despite of the many efforts of prominent scientists, Church's Thesis has never been falsified. There exists a vast literature concerning the thesis. The aim of the book is to provide one volume summary of the state of research on Church's Thesis. These include the following: different formulations of CT, CT and intuitionism, CT and intensional mathematics, CT and physics, the epistemic status of CT, CT and philosophy of mind, provability of CT and CT and functional programming. |
| kleene metamathematics: Reverse Mathematics John Stillwell, 2019-09-24 This volume presents reverse mathematics to a general mathematical audience for the first time. Stillwell gives a representative view of this field, emphasizing basic analysis--finding the right axioms to prove fundamental theorems--and giving a novel approach to logic. to logic. |
| kleene metamathematics: Philosophical Troubles Saul A. Kripke, 2011-12-07 This important new book is the first of a series of volumes collecting the essential articles by the eminent and highly influential philosopher Saul A. Kripke. It presents a mixture of published and unpublished articles from various stages of Kripke's storied career.Included here are seminal and much discussed pieces such as Identity and Necessity, Outline of a Theory of Truth, Speaker's Reference and Semantic Reference, and A Puzzle About Belief. More recent published articles include Russell's Notion of Scope and Frege's Theory of Sense and Reference among others. Several articles are published here for the first time, including both older works (Two Paradoxes of Knowledge, Vacuous Names and Fictional Entities, Nozick on Knowledge) as well as newer (The First Person and Unrestricted Exportation). A Puzzle on Time and Thought was written expressly for this volume.Publication of this volume -- which ranges over epistemology, linguistics, pragmatics, philosophy of language, history of analytic philosophy, theory of truth, and metaphysics -- represents a major event in contemporary analytic philosophy. It will be of great interest to the many who are interested in the work of one its greatest living figures. |
| kleene metamathematics: Martin Davis on Computability, Computational Logic, and Mathematical Foundations Eugenio G. Omodeo, Alberto Policriti, 2017-01-27 This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis’ work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert’s tenth problem. The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis’ work. As a whole, this book shows how Davis’ scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic. |
| kleene metamathematics: Introduction to Mathematical Logic Elliott Mendelson, 2015-05-21 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 Godel, Church, Kleene, Rosse |
| kleene metamathematics: Ontology of Divinity Mirosław Szatkowski, 2024-04-22 This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal’s Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. |
| kleene metamathematics: The Once and Future Turing S. Barry Cooper, Andrew Hodges, 2016-03-24 Alan Turing (1912–1954) made seminal contributions to mathematical logic, computation, computer science, artificial intelligence, cryptography and theoretical biology. In this volume, outstanding scientific thinkers take a fresh look at the great range of Turing's contributions, on how the subjects have developed since his time, and how they might develop still further. The contributors include Martin Davis, J. M. E. Hyland, Andrew R. Booker, Ueli Maurer, Kanti V. Mardia, S. Barry Cooper, Stephen Wolfram, Christof Teuscher, Douglas Richard Hofstadter, Philip K. Maini, Thomas E. Woolley, Eamonn A. Gaffney, Ruth E. Baker, Richard Gordon, Stuart Kauffman, Scott Aaronson, Solomon Feferman, P. D. Welch and Roger Penrose. These specially commissioned essays will provoke and engross the reader who wishes to understand better the lasting significance of one of the twentieth century's deepest thinkers. |
| kleene metamathematics: Fundamental Issues of Artificial Intelligence Vincent C. Müller, 2016-06-07 This volume offers a look at the fundamental issues of present and future AI, especially from cognitive science, computer science, neuroscience and philosophy. This work examines the conditions for artificial intelligence, how these relate to the conditions for intelligence in humans and other natural agents, as well as ethical and societal problems that artificial intelligence raises or will raise. The key issues this volume investigates include the relation of AI and cognitive science, ethics of AI and robotics, brain emulation and simulation, hybrid systems and cyborgs, intelligence and intelligence testing, interactive systems, multi-agent systems, and super intelligence. Based on the 2nd conference on “Theory and Philosophy of Artificial Intelligence” held in Oxford, the volume includes prominent researchers within the field from around the world. |
| kleene metamathematics: Philosophy of Mathematics , 2009-07-08 One of the most striking features of mathematics is the fact that we are much more certain about the mathematical knowledge we have than about what mathematical knowledge is knowledge of. Are numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind? It is these kinds of questions that have encouraged philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics. Over the centuries a number of reasonably well-defined positions about the nature of mathematics have been developed and it is these positions (both historical and current) that are surveyed in the current volume. Traditional theories (Platonism, Aristotelianism, Kantianism), as well as dominant modern theories (logicism, formalism, constructivism, fictionalism, etc.), are all analyzed and evaluated. Leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) is also discussed. The result is a handbook that not only provides a comprehensive overview of recent developments but that also serves as an indispensable resource for anyone wanting to learn about current developments in the philosophy of mathematics.-Comprehensive coverage of all main theories in the philosophy of mathematics-Clearly written expositions of fundamental ideas and concepts-Definitive discussions by leading researchers in the field-Summaries of leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) are also included |
| kleene metamathematics: Foundations of Constructive Mathematics M.J. Beeson, 2012-12-06 This book is about some recent work in a subject usually considered part of logic and the foundations of mathematics, but also having close connec tions with philosophy and computer science. Namely, the creation and study of formal systems for constructive mathematics. The general organization of the book is described in the User's Manual which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, formal systems for constructive mathematics. Con structive mathematics refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind. |
| kleene metamathematics: Notes for Lectures on Metamathematics, Given at Stanford University, 1961-1962 Solomon Feferman, 1962 |
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