Hilbert S Program And The Omega Rule
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Hilbert S Program And The Omega Rule
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Author : Aleksandar Ignjatović
language : en
Publisher:
Release Date : 1992
Hilbert S Program And The Omega Rule written by Aleksandar Ignjatović and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Hilbert algebras categories.
Abstract: "In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of Gödel's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the [omega]-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory Z. On the other hand, he claims that Z with its first order logic is not strong enough to allow a formalization of such an [omega]-rule. This would explain why the unprovability of Con(Z) in Z does not imply that the consistency of Z cannot be proved by finitistic means. We show that Detlefsen's proposal is unacceptable as originally formulated in [1], but that a reasonable modification of the rule he suggests leads to a partial program already studied for many years. We investigate the scope of such a program in terms of proof-theoretic reducibilities. We also show that this partial program encompasses mathematically important theories studied in the 'Reverse Mathematics' program. In order to investigate the provability with such a modified rule, we define new consistency and provability predicates which are weaker than the usual ones. We then investigate their properties, including a few that have no apparent philosophical significance but compare interestingly with the properties of the corresponding standard predicates. Finally, we also discuss some possible extensions of the program, based on the iteration of our [omega]-rule. We determine some of the limitations of such programs, pointing out that these limitations partly explain why partial programs that have been successfully carried out use quite different and substantially more radical extensions of finitistic methods with more general forms of restricted reasoning."
Uncertainty Modeling
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Author : Vladik Kreinovich
language : en
Publisher: Springer
Release Date : 2017-01-31
Uncertainty Modeling written by Vladik Kreinovich and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-31 with Technology & Engineering categories.
This book commemorates the 65th birthday of Dr. Boris Kovalerchuk, and reflects many of the research areas covered by his work. It focuses on data processing under uncertainty, especially fuzzy data processing, when uncertainty comes from the imprecision of expert opinions. The book includes 17 authoritative contributions by leading experts.
Philosophy Of Logic
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Author :
language : en
Publisher: Elsevier
Release Date : 2006-11-29
Philosophy Of Logic written by and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-29 with Mathematics categories.
The papers presented in this volume examine topics of central interest in contemporary philosophy of logic. They include reflections on the nature of logic and its relevance for philosophy today, and explore in depth developments in informal logic and the relation of informal to symbolic logic, mathematical metatheory and the limiting metatheorems, modal logic, many-valued logic, relevance and paraconsistent logic, free logics, extensional v. intensional logics, the logic of fiction, epistemic logic, formal logical and semantic paradoxes, the concept of truth, the formal theory of entailment, objectual and substitutional interpretation of the quantifiers, infinity and domain constraints, the Löwenheim-Skolem theorem and Skolem paradox, vagueness, modal realism v. actualism, counterfactuals and the logic of causation, applications of logic and mathematics to the physical sciences, logically possible worlds and counterpart semantics, and the legacy of Hilbert’s program and logicism. The handbook is meant to be both a compendium of new work in symbolic logic and an authoritative resource for students and researchers, a book to be consulted for specific information about recent developments in logic and to be read with pleasure for its technical acumen and philosophical insights. - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic - Useful bibliographies in every chapter
Truth Proof And Infinity
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Author : P. Fletcher
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Truth Proof And Infinity written by P. Fletcher and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Philosophy categories.
Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms `construction' and `proof' has never been adequately explained (although Kriesel, Goodman and Martin-Löf have attempted axiomatisations). This monograph develops precise (though not wholly formal) definitions of construction and proof, and describes the algorithmic substructure underlying intuitionistic logic. Interpretations of Heyting arithmetic and constructive analysis are given. The philosophical basis of constructivism is explored thoroughly in Part I. The author seeks to answer objections from platonists and to reconcile his position with the central insights of Hilbert's formalism and logic. Audience: Philosophers of mathematics and logicians, both academic and graduate students, particularly those interested in Brouwer and Hilbert; theoretical computer scientists interested in the foundations of functional programming languages and program correctness calculi.
Principia Mathematica
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Author : Alfred North Whitehead
language : en
Publisher: Cambridge University Press
Release Date : 1927
Principia Mathematica written by Alfred North Whitehead and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1927 with Mathematics categories.
The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century.
Recursive Functions And Metamathematics
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Author : Roman Murawski
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
Recursive Functions And Metamathematics written by Roman Murawski and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Philosophy categories.
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.
Unifying The Philosophy Of Truth
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Author : Theodora Achourioti
language : en
Publisher: Springer
Release Date : 2015-06-16
Unifying The Philosophy Of Truth written by Theodora Achourioti and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-16 with Philosophy categories.
This anthology of the very latest research on truth features the work of recognized luminaries in the field, put together following a rigorous refereeing process. Along with an introduction outlining the central issues in the field, it provides a unique and unrivaled view of contemporary work on the nature of truth, with papers selected from key conferences in 2011 such as Truth Be Told (Amsterdam), Truth at Work (Paris), Paradoxes of Truth and Denotation (Barcelona) and Axiomatic Theories of Truth (Oxford). Studying the nature of the concept of ‘truth’ has always been a core role of philosophy, but recent years have been a boom time in the topic. With a wealth of recent conferences examining the subject from various angles, this collection of essays recognizes the pressing need for a volume that brings scholars up to date on the arguments. Offering academics and graduate students alike a much-needed repository of today’s cutting-edge work in this vital topic of philosophy, the volume is required reading for anyone needing to keep abreast of developments, and is certain to act as a catalyst for further innovation and research.
Hilbert S Program
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Author : M. Detlefsen
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Hilbert S Program written by M. Detlefsen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Philosophy categories.
Hilbert's Program was founded on a concern for the phenomenon of paradox in mathematics. To Hilbert, the paradoxes, which are at once both absurd and irresistible, revealed a deep philosophical truth: namely, that there is a discrepancy between the laws accord ing to which the mind of homo mathematicus works, and the laws governing objective mathematical fact. Mathematical epistemology is, therefore, to be seen as a struggle between a mind that naturally works in one way and a reality that works in another. Knowledge occurs when the two cooperate. Conceived in this way, there are two basic alternatives for mathematical epistemology: a skeptical position which maintains either that mind and reality seldom or never come to agreement, or that we have no very reliable way of telling when they do; and a non-skeptical position which holds that there is significant agree ment between mind and reality, and that their potential discrepan cies can be detected, avoided, and thus kept in check. Of these two, Hilbert clearly embraced the latter, and proposed a program designed to vindicate the epistemological riches represented by our natural, if non-literal, ways of thinking. Brouwer, on the other hand, opted for a position closer (in Hilbert's opinion) to that of the skeptic. Having decided that epistemological purity could come only through sacrifice, he turned his back on his classical heritage to accept a higher calling.
Logic S Lost Genius
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Author : Eckart Menzler-Trott
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-05-05
Logic S Lost Genius written by Eckart Menzler-Trott and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-05 with categories.
Gerhard Gentzen (1909–1945) is the founder of modern structural proof theory. His lasting methods, rules, and structures resulted not only in the technical mathematical discipline called “proof theory” but also in verification programs that are essential in computer science. The appearance, clarity, and elegance of Gentzen's work on natural deduction, the sequent calculus, and ordinal proof theory continue to be impressive even today. The present book gives the first comprehensive, detailed, accurate scientific biography expounding the life and work of Gerhard Gentzen, one of our greatest logicians, until his arrest and death in Prague in 1945. Particular emphasis in the book is put on the conditions of scientific research, in this case mathematical logic, in National Socialist Germany, the ideological fight for “German logic”, and their mutual protagonists. Numerous hitherto unpublished sources, family documents, archival material, interviews, and letters, as well as Gentzen's lectures for the mathematical public, make this book an indispensable source of information on this important mathematician, his work, and his time. The volume is completed by two deep substantial essays by Jan von Plato and Craig Smoryński on Gentzen's proof theory; its relation to the ideas of Hilbert, Brouwer, Weyl, and Gödel; and its development up to the present day. Smoryński explains the Hilbert program in more than the usual slogan form and shows why consistency is important. Von Plato shows in detail the benefits of Gentzen's program. This important book is a self-contained starting point for any work on Gentzen and his logic. The book is accessible to a wide audience with different backgrounds and is suitable for general readers, researchers, students, and teachers.
An Introduction To G Del S Theorems
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Author : Peter Smith
language : en
Publisher: Cambridge University Press
Release Date : 2007-07-26
An Introduction To G Del S Theorems written by Peter Smith and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-07-26 with Mathematics categories.
Peter Smith examines Gödel's Theorems, how they were established and why they matter.