The Unprovability Of Consistency
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The Unprovability Of Consistency
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Author : George Boolos
language : en
Publisher: Cambridge University Press
Release Date : 1979-04-30
The Unprovability Of Consistency written by George Boolos 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 1979-04-30 with Mathematics categories.
The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal logic that has persisted for at least thirty years between the followers of C. I. Lewis and W. V. O. Quine. The author employs semantic methods developed by Saul Kripke in his analysis of modal logical systems. The book will be of interest to advanced undergraduate and graduate students in logic, mathematics and philosophy, as well as to specialists in those fields.
The Unprovability Of Consistency
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Author : George Boolos
language : en
Publisher:
Release Date : 1979
The Unprovability Of Consistency written by George Boolos and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1979 with Modality (Logic) categories.
An Introduction To G Del S Theorems
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Author : Peter Smith
language : en
Publisher: Cambridge University Press
Release Date : 2013-02-21
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 2013-02-21 with Biography & Autobiography categories.
A clear and accessible treatment of Gödel's famous, intriguing, but much misunderstood incompleteness theorems, extensively revised in a second edition.
Can Mathematics Be Proved Consistent
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Author : Jan von Plato
language : en
Publisher: Springer Nature
Release Date : 2020-07-24
Can Mathematics Be Proved Consistent written by Jan von Plato and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-07-24 with Mathematics categories.
Kurt Gödel (1906–1978) shook the mathematical world in 1931 by a result that has become an icon of 20th century science: The search for rigour in proving mathematical theorems had led to the formalization of mathematical proofs, to the extent that such proving could be reduced to the application of a few mechanical rules. Gödel showed that whenever the part of mathematics under formalization contains elementary arithmetic, there will be arithmetical statements that should be formally provable but aren’t. The result is known as Gödel’s first incompleteness theorem, so called because there is a second incompleteness result, embodied in his answer to the question "Can mathematics be proved consistent?" This book offers the first examination of Gödel’s preserved notebooks from 1930, written in a long-forgotten German shorthand, that show his way to the results: his first ideas, how they evolved, and how the jewel-like final presentation in his famous publication On formally undecidable propositions was composed.The book also contains the original version of Gödel’s incompleteness article, as handed in for publication with no mentioning of the second incompleteness theorem, as well as six contemporary lectures and seminars Gödel gave between 1931 and 1934 in Austria, Germany, and the United States. The lectures are masterpieces of accessible presentations of deep scientific results, readable even for those without special mathematical training, and published here for the first time.
The Enigma Of Completeness A Journey Through The Labyrinth Of Mathematical Logic
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Author : Pasquale De Marco
language : en
Publisher: Pasquale De Marco
Release Date :
The Enigma Of Completeness A Journey Through The Labyrinth Of Mathematical Logic written by Pasquale De Marco and has been published by Pasquale De Marco this book supported file pdf, txt, epub, kindle and other format this book has been release on with Philosophy categories.
In the realm of mathematics and logic, there exists an enigma that has captivated and challenged the greatest minds for generations: the enigma of completeness. This book embarks on a profound journey to unravel this enigma, venturing into the labyrinth of mathematical logic and exploring the revolutionary implications of Gödel's incompleteness theorems. Kurt Gödel, a visionary mathematician of the 20th century, shattered the long-held belief in the absolute completeness of formal systems with his groundbreaking incompleteness theorems. These theorems revealed that within any axiomatic system, there will always exist true statements that cannot be proven within that system. This startling discovery exposed the inherent limitations of formal logic and raised fundamental questions about the nature of truth, knowledge, and the limits of human reason. This book delves into the profound significance of Gödel's incompleteness theorems, examining their impact on various fields of intellectual inquiry, from philosophy and computer science to artificial intelligence and the nature of reality itself. It explores the quest for alternative foundations of mathematics, the implications for the foundations of computer science, and the potential of artificial intelligence in mathematical discovery. With clarity and rigor, this book unravels the intricate tapestry of mathematical logic, making it accessible to readers from all backgrounds. It illuminates the beauty and elegance of Gödel's theorems, while also exploring their profound implications for our understanding of the universe and our place within it. Join us on this intellectual odyssey as we navigate the labyrinth of completeness, unraveling the mysteries that lie at the heart of mathematical logic. Prepare to be challenged, intrigued, and enlightened as we explore the enigma of completeness, a testament to the boundless capacity of the human mind to grapple with the fundamental questions of existence. This book is an essential read for anyone interested in the foundations of mathematics, logic, philosophy, or the nature of reality. It is a profound exploration of one of the most important intellectual discoveries of the 20th century, and its implications for our understanding of the universe and our place within it. If you like this book, write a review!
On Formally Undecidable Propositions Of Principia Mathematica And Related Systems
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Author : Kurt Gödel
language : en
Publisher: Courier Corporation
Release Date : 2012-05-24
On Formally Undecidable Propositions Of Principia Mathematica And Related Systems written by Kurt Gödel and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-24 with Mathematics categories.
First English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.
Godel S Incompleteness Theorems
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Author : Raymond M. Smullyan
language : en
Publisher: Oxford University Press
Release Date : 1992-08-20
Godel S Incompleteness Theorems written by Raymond M. Smullyan and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-08-20 with Mathematics categories.
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.
Metamathematics Of First Order Arithmetic
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Author : Petr Hájek
language : en
Publisher: Cambridge University Press
Release Date : 2017-03-02
Metamathematics Of First Order Arithmetic written by Petr Hájek 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 2017-03-02 with Mathematics categories.
A much-needed monograph on the metamathematics of first-order arithmetic, paying particular attention to fragments of Peano arithmetic.
John Von Neumann And The Foundations Of Quantum Physics
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Author : Miklós Rédei
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
John Von Neumann And The Foundations Of Quantum Physics written by Miklós Rédei 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 Science categories.
John von Neumann (1903-1957) was undoubtedly one of the scientific geniuses of the 20th century. The main fields to which he contributed include various disciplines of pure and applied mathematics, mathematical and theoretical physics, logic, theoretical computer science, and computer architecture. Von Neumann was also actively involved in politics and science management and he had a major impact on US government decisions during, and especially after, the Second World War. There exist several popular books on his personality and various collections focusing on his achievements in mathematics, computer science, and economy. Strangely enough, to date no detailed appraisal of his seminal contributions to the mathematical foundations of quantum physics has appeared. Von Neumann's theory of measurement and his critique of hidden variables became the touchstone of most debates in the foundations of quantum mechanics. Today, his name also figures most prominently in the mathematically rigorous branches of contemporary quantum mechanics of large systems and quantum field theory. And finally - as one of his last lectures, published in this volume for the first time, shows - he considered the relation of quantum logic and quantum mechanical probability as his most important problem for the second half of the twentieth century. The present volume embraces both historical and systematic analyses of his methodology of mathematical physics, and of the various aspects of his work in the foundations of quantum physics, such as theory of measurement, quantum logic, and quantum mechanical entropy. The volume is rounded off by previously unpublished letters and lectures documenting von Neumann's thinking about quantum theory after his 1932 Mathematical Foundations of Quantum Mechanics. The general part of the Yearbook contains papers emerging from the Institute's annual lecture series and reviews of important publications of philosophy of science and its history.
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 Mathematics 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.