Incompleteness
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Incompleteness
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Author : Rebecca Goldstein
language : en
Publisher: W. W. Norton & Company
Release Date : 2006-01-31
Incompleteness written by Rebecca Goldstein and has been published by W. W. Norton & Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-31 with Biography & Autobiography categories.
"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
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.
Incompleteness The Proof And Paradox Of Kurt G Del Great Discoveries
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Author : Rebecca Goldstein
language : en
Publisher: W. W. Norton & Company
Release Date : 2006-02-17
Incompleteness The Proof And Paradox Of Kurt G Del Great Discoveries written by Rebecca Goldstein and has been published by W. W. Norton & Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-17 with Biography & Autobiography categories.
"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.
Aspects Of Incompleteness
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Author : Per Lindström
language : en
Publisher: Cambridge University Press
Release Date : 2017-03-02
Aspects Of Incompleteness written by Per Lindström 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.
This volume presents some of the main areas and results of general metamathematics, including the results of Gödel et al. on incompleteness.
The Incompleteness Phenomenon
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Author : Martin Goldstern
language : en
Publisher: CRC Press
Release Date : 2018-10-08
The Incompleteness Phenomenon written by Martin Goldstern and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-08 with Mathematics categories.
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.
A Short Remark On G Del Incompleteness Theorem And Its Self Referential Paradox From Neutrosophic Logic Perspective
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Author : V. Christianto
language : en
Publisher: Infinite Study
Release Date :
A Short Remark On G Del Incompleteness Theorem And Its Self Referential Paradox From Neutrosophic Logic Perspective written by V. Christianto and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic.
Incompleteness In The Land Of Sets
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Author : Melvin Fitting
language : en
Publisher:
Release Date : 2007
Incompleteness In The Land Of Sets written by Melvin Fitting and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Incompleteness theorems categories.
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.
There S Something About G Del
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Author : Francesco Berto
language : en
Publisher: John Wiley & Sons
Release Date : 2011-09-13
There S Something About G Del written by Francesco Berto and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-09-13 with Philosophy categories.
Berto's highly readable and lucid guide introduces students and the interested reader to Gödel's celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel's theories Written in an accessible, non-technical style
Information Theoretic Incompleteness
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Author : Gregory J Chaitin
language : en
Publisher: World Scientific
Release Date : 1992-08-24
Information Theoretic Incompleteness written by Gregory J Chaitin and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-08-24 with Computers categories.
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.
Incompleteness For Higher Order Arithmetic
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Author : Yong Cheng
language : en
Publisher: Springer Nature
Release Date : 2019-08-30
Incompleteness For Higher Order Arithmetic written by Yong Cheng and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-08-30 with Mathematics categories.
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.