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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.

Fiction And The Incompleteness Of History

Author : Ying Zhu
ISBN : 3039107461
Genre : Literary Criticism
File Size : 83. 16 MB
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Based on the author's thesis (Doctoral--University of Hong Kong, 2005).

Incompleteness For Higher Order Arithmetic

Author : Yong Cheng
ISBN : 9789811399497
Genre : Mathematics
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Gödel's true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements that are similarly unprovable. A lot of research has since been done in this direction, most notably by Harvey Friedman. A lot of examples of concrete incompleteness with real mathematical content have been found to date. This brief contributes to Harvey Friedman's research program on concrete incompleteness for higher-order arithmetic and gives a specific example of concrete mathematical theorems which is expressible in second-order arithmetic but the minimal system in higher-order arithmetic to prove it is fourth-order arithmetic. This book first examines the following foundational question: are all theorems in classic mathematics expressible in second-order arithmetic provable in second-order arithmetic? The author gives a counterexample for this question and isolates this counterexample from the Martin-Harrington Theorem in set theory. It shows that the statement “Harrington's principle implies zero sharp" is not provable in second-order arithmetic. This book further examines what is the minimal system in higher-order arithmetic to prove the theorem “Harrington's principle implies zero sharp" and shows that it is neither provable in second-order arithmetic or third-order arithmetic, but provable in fourth-order arithmetic. The book also examines the large cardinal strength of Harrington's principle and its strengthening over second-order arithmetic and third-order arithmetic.

The Incompleteness Phenomenon

Author : Martin Goldstern
ISBN : 9781439863534
Genre : Mathematics
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This introduction to mathematical logic takes Gödel's incompleteness theorem as a starting point. It goes beyond a standard text book and should interest everyone from mathematicians to philosophers and general readers who wish to understand the foundations and limitations of modern mathematics.

Information Randomness Incompleteness

Author : Gregory J. Chaitin
ISBN : 9810201710
Genre : Mathematics
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This book contains in easily accessible form all the main ideas of the creator and principal architect of algorithmic information theory. This expanded second edition has added thirteen abstracts, a 1988 Scientific American Article, a transcript of a EUROPALIA 89 lecture, an essay on biology, and an extensive bibliography. Its new larger format makes it easier to read. Chaitin's ideas are a fundamental extension of those of G”del and Turning and have exploded some basic assumptions of mathematics and thrown new light on the scientific method, epistemology, probability theory, and of course computer science and information theory.

Incompleteness The Proof And Paradox Of Kurt G?del Great Discoveries

Author : Rebecca Goldstein
ISBN : 9780393327601
Genre : Biography & Autobiography
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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.

Information Theoretic Incompleteness

Author : Gregory J. Chaitin
ISBN : 9810236956
Genre : Computers
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In this mathematical autobiography, Gregory Chaitin presents a technical survey of his work and a nontechnical discussion of its significance. The volume is an essential companion to the earlier collection of Chaitin's papers Information, Randomness and Incompleteness, also published by World Scientific.The technical survey contains many new results, including a detailed discussion of LISP program size and new versions of Chaitin's most fundamental information-theoretic incompleteness theorems. The nontechnical part includes the lecture given by Chaitin in G?del's classroom at the University of Vienna, a transcript of a BBC TV interview, and articles from New Scientist, La Recherche, and the Mathematical Intelligencer.

Incompleteness A Mark Of True Work A Commemoration Sermon On Matt X 23 Etc

Author : Charles John VAUGHAN (Dean of Llandaff.)
ISBN : BL:A0021774631
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Belief And Incompleteness

Author : Kurt Konolige
ISBN : STANFORD:36105039643494
Genre : Analysis (Philosophy)
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Both agents are state-of-theart constructions, incorporating the latest Al research in chess playing, natural-language understanding, planning, etc. But because of the overwhelming combinatorics of' chess, neither they nor the fastest foreseeable computers would be able to search the entire game tree to find out whether White has a forced win. Why then do they come to such an odd conclusion about their own knowledge of the game? The chess scenario is an anecdotal example of the way inaccurate cognitive models can lead to behavior that is less than intelligent in artificial agents. In this case, the agents' model of belief is not correct. They make the assumption that an agent actually knows all the consequences of his beliefs. S1 knows that chess is a finite game, and thus reasons that, in principle, knowing the rules of chess is all that is required to figure out whether White has a forced ini%ial win. Mter learning that S2 does indeed know the rules of chess he comes to the erroneous conclusion that S2 also knows this particular consequence of the rules. And S2 himself, reflecting on his own knowledge in the same manner, arrives at the same conclusion, even though in actual fact he could never carry out the computations necessary to demonstrate it.

G?del S Incompleteness Theorem

Author : V. A. Uspensky
ISBN : PSU:000014514777
Genre : Gödel's theorem
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Interfirm Partial Ownership Under Contractual Incompleteness

Author : Sudipto Dasgupta
ISBN : UCSD:31822021229109
Genre :
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Contractual Incompleteness And The Nature Of Market Interactions

Author : Martin Brown
ISBN : UVA:X004529452
Genre : Incomplete contracts
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Mathematical Methods In Linguistics

Author : Barbara B.H. Partee
ISBN : 9027722455
Genre : Language Arts & Disciplines
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Elementary set theory accustoms the students to mathematical abstraction, includes the standard constructions of relations, functions, and orderings, and leads to a discussion of the various orders of infinity. The material on logic covers not only the standard statement logic and first-order predicate logic but includes an introduction to formal systems, axiomatization, and model theory. The section on algebra is presented with an emphasis on lattices as well as Boolean and Heyting algebras. Background for recent research in natural language semantics includes sections on lambda-abstraction and generalized quantifiers. Chapters on automata theory and formal languages contain a discussion of languages between context-free and context-sensitive and form the background for much current work in syntactic theory and computational linguistics. The many exercises not only reinforce basic skills but offer an entry to linguistic applications of mathematical concepts. For upper-level undergraduate students and graduate students in theoretical linguistics, computer-science students with interests in computational linguistics, logic programming and artificial intelligence, mathematicians and logicians with interests in linguistics and the semantics of natural language.

Metamathematics Machines And G?del S Proof

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

Dialogues In Phenomenology

Author : D. Ihde
ISBN : 9789401016155
Genre : Philosophy
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Phenomenology in the United States is in a state of ferment and change. Not all the changes are happy ones, however, for some of the most prominent philosophers of the first generation of phenomenologists have died: in 1959 Alfred Schutz, and within the past two years John \Vild, Dorion Cairns, and Aron Gur witsch. These thinkers, though often confronting a hostile intel lectual climate, were nevertheless persistent and profoundly influential-through their own works, and through their students. The two sources associated with their names, The Graduate Faculty of The New School for Social Research, and the circle around John Wild first at Harvard and later at Northwestern and Yale, produced a sizable portion of the now second gener ation American phenomenological philosophers. In a way, it was the very hostility of the American philo sophical milieu which became an important factor in the ferment now taking place. Although the older, first generation phenome nologists were deeply conversant with other philosophical move ments here and abroad, their efforts at meaningful dialogue were largely ignored. Determined not to remain isolated from the dominant currents of Anglo-American philosophy in par ticular, the second generation opened the way to a dialogue with analytic philosophers, especially through the efforts of the Society for Phenomenology and Existential Philosophy, led by 2 INTRODUCTION such men as James M. Edie and Hubert Dreyfus and, in other respects, Herbert Spiegelberg and Maurice Natanson.

Precise Conclusions Under Uncertainty And Incompleteness In Deductive Database Systems

Author : Helmut Thöne
ISBN : OCLC:311941248
Genre :
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Flexible And Efficient Information Handling

Author : David Bell
ISBN : 9783540359692
Genre : Computers
File Size : 77. 79 MB
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This book constitutes the refereed proceedings of the 23rd British National Conference on Databases, BNCOD 23, held in Belfast, Northern Ireland, UK in July 2006. The 12 revised full papers and 6 revised short papers presented together with 2 invited lectures and 13 poster papers were carefully reviewed and selected from 58 submissions. The papers are organized in topical sections on data modelling and architectures and transaction management, data integration and interoperability and information retrieval, query processing and optimisation, data mining, data warehousing and decision-support systems, as well as data streaming.

The Formal Semantics Of Programming Languages

Author : Glynn Winskel
ISBN : 0262731037
Genre : Computers
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The Formal Semantics of Programming Languages provides the basic mathematical techniques necessary for those who are beginning a study of the semantics and logics of programming languages. These techniques will allow students to invent, formalize, and justify rules with which to reason about a variety of programming languages. Although the treatment is elementary, several of the topics covered are drawn from recent research, including the vital area of concurency. The book contains many exercises ranging from simple to miniprojects.Starting with basic set theory, structural operational semantics is introduced as a way to define the meaning of programming languages along with associated proof techniques. Denotational and axiomatic semantics are illustrated on a simple language of while-programs, and fall proofs are given of the equivalence of the operational and denotational semantics and soundness and relative completeness of the axiomatic semantics. A proof of Godel's incompleteness theorem, which emphasizes the impossibility of achieving a fully complete axiomatic semantics, is included. It is supported by an appendix providing an introduction to the theory of computability based on while-programs. Following a presentation of domain theory, the semantics and methods of proof for several functional languages are treated. The simplest language is that of recursion equations with both call-by-value and call-by-name evaluation. This work is extended to lan guages with higher and recursive types, including a treatment of the eager and lazy lambda-calculi. Throughout, the relationship between denotational and operational semantics is stressed, and the proofs of the correspondence between the operation and denotational semantics are provided. The treatment of recursive types - one of the more advanced parts of the book - relies on the use of information systems to represent domains. The book concludes with a chapter on parallel programming languages, accompanied by a discussion of methods for specifying and verifying nondeterministic and parallel programs.

G?del Putnam And Functionalism

Author : Jeff Buechner
ISBN : 9780262261975
Genre : Philosophy
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The first systematic examination of Hilary Putnam's arguments against computational functionalism challenges each of Putnam's main arguments. With mind-brain identity theories no longer dominant in philosophy of mind in the late 1950s, scientific materialists turned to functionalism, the view that the identity of any mental state depends on its function in the cognitive system of which it is a part. The philosopher Hilary Putnam was one of the primary architects of functionalism and was the first to propose computational functionalism, which views the human mind as a computer or an information processor. But, in the early 1970s, Putnam began to have doubts about functionalism, and in his masterwork Representation and Reality (MIT Press, 1988), he advanced four powerful arguments against his own doctrine of computational functionalism. In Gödel, Putnam, and Functionalism, Jeff Buechner systematically examines Putnam's arguments against functionalism and contends that they are unsuccessful. Putnam's first argument uses Gödel's incompleteness theorem to refute the view that there is a computational description of human reasoning and rationality; his second, the “triviality argument,” demonstrates that any computational description can be attributed to any physical system; his third, the multirealization argument, shows that there are infinitely many computational realizations of an arbitrary intentional state; his fourth argument buttresses this assertion by showing that there cannot be local computational reductions because there is no computable partitioning of the infinity of computational realizations of an arbitrary intentional state into a single package or small set of packages (equivalence classes). Buechner analyzes these arguments and the important inferential connections among them—for example, the use of both the Gödel and triviality arguments in the argument against local computational reductions—and argues that none of Putnam's four arguments succeeds in refuting functionalism. Gödel, Putnam, and Functionalism will inspire renewed discussion of Putnam's influential book and will confirm Representation and Reality as a major work by a major philosopher.

I Know What I Like

Author : Lawrence Alena A.
ISBN : OCLC:1097320473
Genre :
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