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Incompleteness
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Book Synopsis Incompleteness by : Rebecca Goldstein
Download or read book Incompleteness written by Rebecca Goldstein and published by W. W. Norton & Company. This book was released on 2006-01-31 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: "An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
Book Synopsis Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries) by : Rebecca Goldstein
Download or read book Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries) written by Rebecca Goldstein and published by W. W. Norton & Company. This book was released on 2006-02-17 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: "A gem…An unforgettable account of one of the great moments in the history of human thought." —Steven Pinker Probing the life and work of Kurt Gödel, Incompleteness indelibly portrays the tortured genius whose vision rocked the stability of mathematical reasoning—and brought him to the edge of madness.
Book Synopsis Incompleteness by : Rebecca Goldstein
Download or read book Incompleteness written by Rebecca Goldstein and published by W. W. Norton & Company. This book was released on 2005 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Considered the 20th century's greatest mathematician, Kurt Godel is the subject of this lucid and accessible study, which explains the significance of his theorems and the remarkable vision behind them, while bringing this eccentric, tortured genius and his world to life.
Book Synopsis The Incompleteness Phenomenon by : Martin Goldstern
Download or read book The Incompleteness Phenomenon written by Martin Goldstern and published by CRC Press. This book was released on 2018-10-08 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Godel's Incompleteness Theorems by : Raymond M. Smullyan
Download or read book Godel's Incompleteness Theorems written by Raymond M. Smullyan and published by Oxford University Press. This book was released on 1992-08-20 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Incompleteness in the Land of Sets by : Melvin Fitting
Download or read book Incompleteness in the Land of Sets written by Melvin Fitting and published by . This book was released on 2007 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Russell's paradox arises when we consider those sets that do not belong to themselves. The collection of such sets cannot constitute a set. Step back a bit. Logical formulas define sets (in a standard model). Formulas, being mathematical objects, can be thought of as sets themselves-mathematics reduces to set theory. Consider those formulas that do not belong to the set they define. The collection of such formulas is not definable by a formula, by the same argument that Russell used. This quickly gives Tarski's result on the undefinability of truth. Variations on the same idea yield the famous results of Gödel, Church, Rosser, and Post. This book gives a full presentation of the basic incompleteness and undecidability theorems of mathematical logic in the framework of set theory. Corresponding results for arithmetic follow easily, and are also given. Gödel numbering is generally avoided, except when an explicit connection is made between set theory and arithmetic. The book assumes little technical background from the reader. One needs mathematical ability, a general familiarity with formal logic, and an understanding of the completeness theorem, though not its proof. All else is developed and formally proved, from Tarski's Theorem to Gödel's Second Incompleteness Theorem. Exercises are scattered throughout.
Book Synopsis Fiction and the Incompleteness of History by : Zhu Ying
Download or read book Fiction and the Incompleteness of History written by Zhu Ying and published by Peter Lang. This book was released on 2006 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the author's thesis (Doctoral--University of Hong Kong, 2005).
Book Synopsis Incompleteness for Higher-Order Arithmetic by : Yong Cheng
Download or read book Incompleteness for Higher-Order Arithmetic written by Yong Cheng and published by Springer Nature. This book was released on 2019-08-30 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Incompleteness Mobility and Conviviality by : Francis Nyamnjoh
Download or read book Incompleteness Mobility and Conviviality written by Francis Nyamnjoh and published by African Books Collective. This book was released on 2023-12-05 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: Central to the Jensen Memorial Lectures 2023 is an invitation to take incompleteness seriously in how we imagine, relate to and seek to understand a world in perpetual motion. Despite our instinct for and obsession with completeness, we are constantly reminded that the sooner one recognises and provides for incompleteness and the conviviality it inspires as the normal way of being, the better we are for it. Fluidity, compositeness and the capacity to be present in multiple places and forms simultaneously in whole or in fragments are core characteristics of reality and ontology of incompleteness. How would we frame our curiosities and conversations about processes, relationships and phenomena with an understanding of the universality of incompleteness and mobility? West and Central Africa, for example, are regions where it is commonplace to embrace and celebrate incompleteness in nature, the suprasensory, human beings, human actions, human inventions and human achievements. The lectures indicate how we could draw inspiration in this regard to inform current clamours for decolonisation and the growing ambivalence about rapid advances in digital technologies (artificial intelligence (AI) in particular), as well as with twenty-first century concerns about migrants and strangers knocking at the doors of opportunities we feel more entitled to as bona fide citizens and insiders. The lectures draw on the writings of Amos Tutuola as well as from popular ideas of personhood and agency in Africa, to make a case for sidestepped and silenced traditions of knowledge. They highlight Africa’s possibilities, prospects and emergent capacities for being and becoming in tune with the continent’s creativity and imagination. They speak to the nimble-footed flexible-minded frontier African at the crossroads and junctions of myriad encounters, facilitating creative conversations and challenging regressive logics of exclusionary claims and articulation of identities and achievements. The traditions of knowledge discussed in these lectures do not only speak to Africans, but to the world, as the philosophies explored have universal application. “The crucial anthropological question of relationality and othering is at the heart of this original and enlightening book. Nyamnjoh cautions the missionaries of decoloniality against the risk of substituting one illusion of completeness with another. For him, incompleteness is the basis of any healthy exchange. He therefore recommends embracing the universality of incompleteness in motion and taking seriously an ancestral tradition of self-extension through creative imagination in this anxious age of artificial intelligence. Forcefully argued and abundantly substantiated – with finesse and laughter that run through it – this book will be a milestone by making us rediscover the demands and the magic of fieldwork.” Prof. Dr. Mamadou Diawara, Goethe University, Frankfurt/Main Frobenius-Institut, Frankfurt/Main Point Sud, Bamako, Mali
Book Synopsis Gödel's Incompleteness Theorems by : Juliette Kennedy
Download or read book Gödel's Incompleteness Theorems written by Juliette Kennedy and published by Cambridge University Press. This book was released on 2022-04-14 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
Book Synopsis Aspects of Incompleteness by : Per Lindström
Download or read book Aspects of Incompleteness written by Per Lindström and published by Cambridge University Press. This book was released on 2017-03-02 with total page 143 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents some of the main areas and results of general metamathematics, including the results of Gödel et al. on incompleteness.
Book Synopsis Incompleteness At Every Turn by : S. Rishabh Bhatt
Download or read book Incompleteness At Every Turn written by S. Rishabh Bhatt and published by S. Rishabh Bhatt. This book was released on 2024-02-13 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Incompleteness At Every Turn" This book is a journey of incompleteness in which every story is wandering somewhere in search of its completeness. Ask yourself the answer to each story and see what answer your heart gives. I believe you will always get a positive answer. Whenever negativity develops inside us, our desires overpower us at that moment. We are not able to understand the difference between right and wrong. But in the meantime, if we find a small pitiful scene, then the unit of positivity within us gets awakened. Believe me, this unit of positivity is the identity of this human life. We have to fight the incompleteness within us and become thirsty and search for the well. If we reach near this well, our thirst will be quenched for a few moments by the water of that well. Life also has the same relation with incompleteness. We try to complete what we know is incomplete. After it is complete, some more incompleteness become visible. Just like drinking water from a well once quenches thirst only for a few moments. After some time, we feel thirsty again and we need water again. This same cycle continues in our lives between incompleteness and life. Therefore, we should always keep moving forward while fighting this.
Book Synopsis Incompleteness: Donald Trump, Populism and Citizenship by : B. Nyamnjoh
Download or read book Incompleteness: Donald Trump, Populism and Citizenship written by B. Nyamnjoh and published by African Books Collective. This book was released on 2022-01-01 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a study of how Donald J. Trump, his populist credentials notwithstanding, borrows without acknowledgment and stubbornly refuses to come to terms with his indebtedness. Taken together with mobility and conviviality, the principle of incompleteness enables us to distinguish between inclusionary and exclusionary forms of populism, and when it is fuelled by ambitions of superiority and zero-sum games of conquest.
Book Synopsis Information-theoretic Incompleteness by : Gregory J Chaitin
Download or read book Information-theoretic Incompleteness written by Gregory J Chaitin and published by World Scientific. This book was released on 1992-08-24 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Gšdel'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.
Book Synopsis Systemics of Incompleteness and Quasi-Systems by : Gianfranco Minati
Download or read book Systemics of Incompleteness and Quasi-Systems written by Gianfranco Minati and published by Springer. This book was released on 2019-06-20 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the proceedings of the Seventh National Conference of the Italian Systems Society. The title, Systemics of Incompleteness and Quasi-Systems, aims to underline the need for Systemics and Systems Science to deal with the concepts of incompleteness and quasiness. Classical models of Systemics are intended to represent comprehensive aspects of phenomena and processes. They consider the phenomena in their temporal and spatial completeness. In these cases, possible incompleteness in the modelling is assumed to have a provisional or practical nature, which is still under study, and because there is no theoretical reason why the modelling cannot be complete. In principle, this is a matter of non-complex phenomena, to be considered using the concepts of the First Systemics. When dealing with emergence, there are phenomena which must be modelled by systems having multiple models, depending on the aspects being taken into consideration. Here, incompleteness in the modelling is intrinsic, theoretically relating changes in properties, structures, and status of system. Rather than consider the same system parametrically changing over time, we consider sequences of systems coherently. We consider contexts and processes for which modelling is incomplete, being related to only some properties, as well as those for which such modelling is theoretically incomplete—as in the case of processes of emergence and for approaches considered by the Second Systemics. In this regard, we consider here the generic concept of quasi explicating such incompleteness. The concept of quasi is used in various disciplines including quasi-crystals, quasi-particles, quasi-electric fields, and quasi-periodicity. In general, the concept of quasiness for systems concerns their continuous structural changes which are always meta-stable, waiting for events to collapse over other configurations and possible forms of stability; whose equivalence depends on the type of phenomenon under study. Interest in the concept of quasiness is not related to its meaning of rough approximation, but because it indicates an incompleteness which is structurally sufficient to accommodate processes of emergence and sustain coherence or generate new, equivalent or non-equivalent, levels. The conference was devoted to identifying, discussing and understanding possible interrelationships of theoretical disciplinary improvements, recognised as having prospective fundamental roles for a new Quasi-Systemics. The latter should be able to deal with problems related to complexity in more general and realistic ways, when a system is not always a system and not always the same system. In this context, the inter-disciplinarity should consist, for instance, of a constructionist, incomplete, non-ideological, multiple, contradiction-tolerant, Systemics, always in progress, and in its turn, emergent.
Book Synopsis Information, Randomness & Incompleteness by : Gregory J. Chaitin
Download or read book Information, Randomness & Incompleteness written by Gregory J. Chaitin and published by World Scientific. This book was released on 1990-01-01 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Gdel 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.
Book Synopsis Incompleteness and Uncertainty in Information Systems by : V.S. Alagar
Download or read book Incompleteness and Uncertainty in Information Systems written by V.S. Alagar and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Software Engineering and Knowledgebase Systems (SOFfEKS) Research Group of the Department of Computer Science, Concordia University, Canada, organized a workshop on Incompleteness and Uncertainty in Information Systems from October 8-9, 1993 in Montreal. A major aim of the workshop was to bring together researchers who share a concern for issues of incompleteness and uncertainty. The workshop attracted people doing fundamental research and industry oriented research in databases, software engineering and AI from North America, Europe and Asia. The workshop program featured six invited talks and twenty other presentations. The invited speakers were: Martin Feather (University of Southern CalifornialInformation Systems Institute) Laks V. S. Lakshmanan (Concordia University) Ewa Orlowska (Polish Academy of Sciences) z. Pawlak (Warsaw Technical University and Academy of Sciences) F. Sadri (Concordia University) A. Skowron (Warsaw University) The papers can be classified into four groups: rough sets and logic, concept analysis, databases and information retrieval, and software engineering. The workshop opened with a warm welcome speech from Dr. Dan Taddeo, Dean, Faculty of Engineering and Computer Science. The first day's presentations were in rough sets, databases and information retrieval. Papers given on the second day centered around software engineering and concept analysis. Sufficient time was given in between presentations to promote active interactions and numerous lively discussions. At the end of two days, the participants expressed their hope that this workshop would be continued.
