Godels Proof

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Godel S Proof

Author : Ernest Nagel
ISBN : 9780359079261
Genre :
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From the Introduction. In 1931 there appeared in a German scientific periodical a relatively short paper with the forbidding title ""Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme"" (""On Formally Undecidable propositions of Principia Mathematica and Related Systems""). Its author was Kurt Godel, then a young mathematician of 25 at the University of Vienna and since 1938 a permanent member of the Institute for Advanced Study at Princeton. The paper is a milestone in the history of logic and mathematics. When Harvard University awarded Godel an honorary degree in 1952, the citation described the work as one of the most important advances in logic in modern times. At the time of its appearance, however, neither the title of Godel's paper nor its content was intelligible to most mathematicians.
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G Del S Proof

Author : Ernest Nagel
ISBN : 9780415355285
Genre : Philosophy
File Size : 77.13 MB
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'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'– The Guardian In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, Godel’s Proofby Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. Kurt Godel(1906 – 1978) Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J.
Category: Philosophy

Godel S Incompleteness Theorems

Author : Raymond M. Smullyan
ISBN : 9780195364378
Genre : Mathematics
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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.
Category: Mathematics

Kurt G Del Collected Works Volume Iii

Author : Kurt Gödel
ISBN : 0195072553
Genre : Mathematics
File Size : 32.34 MB
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Kurt Gödel was the greatest logician of this century. This third volume of his collected works consists of previously unpublished material, both essays and lectures.
Category: Mathematics

Metamathematics Machines And G Del S Proof

Author : N. Shankar
ISBN : 0521585333
Genre : Computers
File Size : 22.46 MB
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Describes the use of computer programs to check several proofs in the foundations of mathematics.
Category: Computers

Incompleteness The Proof And Paradox Of Kurt G Del Great Discoveries

Author : Rebecca Goldstein
ISBN : 9780393327601
Genre : Biography & Autobiography
File Size : 72.78 MB
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A portrait of the eminent twentieth-century mathematician discusses his theorem of incompleteness, relationships with such contemporaries as Albert Einstein, and untimely death as a result of mental instability and self-starvation.
Category: Biography & Autobiography

G Del S Theorem In Focus

Author : Stuart Shanker
ISBN : 9780415045759
Genre : Mathematics
File Size : 37.40 MB
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Beginning with a short biography of Kurt Gödel, 'Gödel's Theorem in Focus' provides the reader with a clear guide to the mechanics of Gödel's proof in a format intelligible to the non-mathematician. The book moves on to explanations of the mechanics of Gödel's proof and its significance for mathematical logic and the philosophy of mathematics. In the final section, S. G. Shanker presents a major new critique of Gödel's theorem.
Category: Mathematics

Internal Logic

Author : Y. Gauthier
ISBN : 9789401700832
Genre : Mathematics
File Size : 23.85 MB
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Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene.
Category: Mathematics

Consciousness And The Brain

Author : Gordon Globus
ISBN : 9781468421965
Genre : Psychology
File Size : 79.54 MB
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The relationship of consciousness to brain, which Schopenhauer grandly referred to as the "world knot," remains an unsolved problem within both philosophy and science. The central focus in what follows is the relevance of science---from psychoanalysis to neurophysiology and quantum physics-to the mind-brain puzzle. Many would argue that we have advanced little since the age of the Greek philosophers, and that the extraordinary accumulation of neuroscientific knowledge in this century has helped not at all. Increas ingly, philosophers and scientists have tended to go their separate ways in considering the issues, since they tend to differ in the questions that they ask, the data and ideas which are provided for consideration, their methods for answering these questions, and criteria for judging the acceptability of an answer. But it is our conviction that philosophers and scientists can usefully interchange, at least to the extent that they provide co~straints upon each other's preferred strategies, and it may prove possible for more substantive progress to be made. Philosophers have said some rather naive things by ignoring the extraordinary advances in the neurosciences in the twentieth century. The skull is not filled with green cheese! On the other hand, the arrogance of many scientists toward philosophy and their faith in the scientific method is equally naive. Scientists clearly have much to learn from philosophy as an intellectual discipline.
Category: Psychology

The Computational Beauty Of Nature

Author : Gary William Flake
ISBN : 0262561271
Genre : Computers
File Size : 38.97 MB
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In this book, Gary William Flake develops in depth the simple idea that recurrent rules can produce rich and complicated behaviors. Distinguishing "agents" (e.g., molecules, cells, animals, and species) from their interactions (e.g., chemical reactions, immune system responses, sexual reproduction, and evolution), Flake argues that it is the computational properties of interactions that account for much of what we think of as "beautiful" and "interesting." From this basic thesis, Flake explores what he considers to be today's four most interesting computational topics: fractals, chaos, complex systems, and adaptation.
Category: Computers

The Shorter Routledge Encyclopedia Of Philosophy

Author : Edward Craig
ISBN : 9781134344086
Genre : Philosophy
File Size : 38.8 MB
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The Shorter REP presents the very best of the acclaimed ten volume Routledge Encyclopedia of Philosophy in a single volume. It makes a selection of the most important entries available for the first time and covers all you need to know about philosophy, from Aristotle to Wittgenstein and animals and ethics to scientific method. Comprising over 900 entries and covering the major philosophers and philosophical topics, The Shorter REP includes the following special features: Unrivalled coverage of major philosophers, themes, movements and periods making the volume indispensable for any student or general reader Fully cross-referenced Revised versions of many of the most important entries, including fresh suggestions for further reading Over twenty brand new entries on important new topics such as Cloning and Sustainability entries by many leading philosophers such as Bernard Williams, Martha Nussbaum, Richard Rorty, Onora O'Neill, T.M. Scanlon and Anthony Appiah Striking new text design to help locate key entries quickly and easily An outstanding guide to all things philosophical, The Shorter Routledge Encyclopedia of Philosophy provides an unrivalled introduction to the subject for students and general readers alike.
Category: Philosophy

Where Is The G Del Point Hiding Gentzen S Consistency Proof Of 1936 And His Representation Of Constructive Ordinals

Author : Anna Horská
ISBN : 9783319021713
Genre : Philosophy
File Size : 28.11 MB
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This book explains the first published consistency proof of PA. It contains the original Gentzen's proof, but it uses modern terminology and examples to illustrate the essential notions. The author comments on Gentzen's steps which are supplemented with exact calculations and parts of formal derivations. A notable aspect of the proof is the representation of ordinal numbers that was developed by Gentzen. This representation is analysed and connection to set-theoretical representation is found, namely an algorithm for translating Gentzen's notation into Cantor normal form. The topic should interest researchers and students who work on proof theory, history of proof theory or Hilbert's program and who do not mind reading mathematical texts.​
Category: Philosophy

Set Theory And Its Philosophy

Author : Michael Potter
ISBN : 9780191556432
Genre : Philosophy
File Size : 50.85 MB
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Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.
Category: Philosophy

Trends In Computer Science Engineering And Information Technology

Author : Dhinaharan Nagamalai
ISBN : 9783642240423
Genre : Computers
File Size : 86.88 MB
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This book constitutes the refereed proceedings of the First International Conference on Computer Science, Engineering and Information Technology, CCSEIT 2011, held in Tirunelveli, India, in September 2011. The 73 revised full papers were carefully reviewed and selected from more than 400 initial submissions. The papers feature significant contributions to all major fields of the Computer Science and Information Technology in theoretical and practical aspects.
Category: Computers

Mathematics And The Roots Of Postmodern Thought

Author : Vladimir Tasi¢c
ISBN : 0195139674
Genre : Mathematics
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1. Introduction. 2. Around the Cartesian Circuit. 2.1. Imagination. 2.2. Intuition. 2.3. Counting to One. 3. Space Oddity and Linguistic Turn. 4. Wound of Language. 4.1. Being and Time Continuum. 4.2. Language and Will. 5. Beyond the Code. 5.1. Medium of Free Becoming. 5.2. Nonpresence of Identity. 6. The Expired Subject. 6.1. Empire of Signs. 6.2. Mechanical Bride. 7. The Vanishing Author. 8. Say Hello to the Structure Bubble. 8.1. Algebra of Language. 8.2. Functionalism Chic. 9. Don't Think, Look. 9.1. Interpolating the Self. 9.2. Language Games. 9.3. Thermostats "R" Us. 10. Postmo.
Category: Mathematics

The P Np Question And G Del S Lost Letter

Author : Richard J. Lipton
ISBN : 1441971556
Genre : Computers
File Size : 76.47 MB
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? DoesP=NP. In just ?ve symbols Dick Karp –in 1972–captured one of the deepest and most important questions of all time. When he ?rst wrote his famous paper, I think it’s fair to say he did not know the depth and importance of his question. Now over three decades later, we know P=NP is central to our understanding of compu- tion, it is a very hard problem, and its resolution will have potentially tremendous consequences. This book is a collection of some of the most popular posts from my blog— Godel ̈ Lost Letter andP=NP—which I started in early 2009. The main thrust of the blog, especially when I started, was to explore various aspects of computational complexity around the famousP=NP question. As I published posts I branched out and covered additional material, sometimes a timely event, sometimes a fun idea, sometimes a new result, and sometimes an old result. I have always tried to make the posts readable by a wide audience, and I believe I have succeeded in doing this.
Category: Computers

Darwin S Dangerous Idea

Author : Daniel C. Dennett
ISBN : 9781439126295
Genre : Science
File Size : 68.27 MB
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In a book that is both groundbreaking and accessible, Daniel C. Dennett, whom Chet Raymo of The Boston Globe calls "one of the most provocative thinkers on the planet," focuses his unerringly logical mind on the theory of natural selection, showing how Darwin's great idea transforms and illuminates our traditional view of humanity's place in the universe. Dennett vividly describes the theory itself and then extends Darwin's vision with impeccable arguments to their often surprising conclusions, challenging the views of some of the most famous scientists of our day.
Category: Science

Formal Logic

Author : Richard C. Jeffrey
ISBN : 0872208133
Genre : Philosophy
File Size : 77.4 MB
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The first beginning logic text to employ the tree method—a complete formal system of first-order logic that is remarkably easy to understand and use—this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems.This new edition provides additional problems, solutions to selected problems, and two new Supplements: “Truth-Functional Equivalence” reinstates material on that topic from the second edition that was omitted in the third, and “Variant Methods, in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification—once contemplated by Jeffrey—can result in a dramatic speed-up of certain proofs.
Category: Philosophy

The Search For The Meaning Of Space Time And Matter

Author : Kai Woehler
ISBN : 9781469101989
Genre : Science
File Size : 26.48 MB
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The book gives a comprehensive introduction for interested general readers, into the development and structure of concepts, ideas and theory formation about the elementary building blocks of matter, the forces with which these particles interact and about the fundamental nature of space itself. Einsteins theory of the cosmos and the recent discovery of the presence of a dark energy which leads to an accelerated expansion of cosmic space, provide the background for the most astonishing recent developments in the search for the origin of space and matter. The String-Theory revolution has led to the notion that nature may not follow one unique set of laws to build worlds, but that innumerable many possible universes may exist, that worlds may be emerging and disappearing like biological species and that our existence may be extraordinarily rare and therefore precious. An introduction to the concept of emergence in self-organizing systems is given to make the connection to the idea that Emergence may be the inherent creative property of space and matter at the quantum level.
Category: Science