Why The Cantor Diagonal Argument Is Not Valid
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Why The Cantor Diagonal Argument Is Not Valid
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Author : Pravin Johri
language : en
Publisher: Createspace Independent Publishing Platform
Release Date : 2018-06-10
Why The Cantor Diagonal Argument Is Not Valid written by Pravin Johri and has been published by Createspace Independent Publishing Platform this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-10 with categories.
The Cantor Diagonal Argument (CDA) is the quintessential result in Cantor's infinite set theory. This is one procedure that almost everyone who studies this subject finds astounding. However, mathematicians maintain that the CDA is absolutely correct and that the "countless" people trying to repudiate the CDA are not only wrong but are seemingly "irrational" enough to challenge such a widely accepted result.This book outlines all the different issues with the CDA. And, there are many.This book does not attempt to disprove the CDA by finding fault with it. Since the mathematical community has not bought into any of the tens of counterarguments it likely will ignore yet one more.Instead, assuming the CDA is correct, we create a situation where the CDA produces results when it really shouldn't and use the CDA itself to discredit the CDA.Cantor's infinite set theory is largely based on arbitrary rules, confounding axioms, and logic that defies intuition and common sense. Our previous books explain exactly what is wrong and why. The theory is hopelessly flawed because the starting assumption - the axiom of infinity - is wrong. There is no such thing as an infinite set. But mathematicians stubbornly stick to their belief that everything is correct.Hopefully this is the straw that breaks the camel's back!
People Problems And Proofs
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Author : Richard J. Lipton
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-11
People Problems And Proofs written by Richard J. Lipton 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 2013-12-11 with Computers categories.
People, problems, and proofs are the lifeblood of theoretical computer science. Behind the computing devices and applications that have transformed our lives are clever algorithms, and for every worthwhile algorithm there is a problem that it solves and a proof that it works. Before this proof there was an open problem: can one create an efficient algorithm to solve the computational problem? And, finally, behind these questions are the people who are excited about these fundamental issues in our computational world. In this book the authors draw on their outstanding research and teaching experience to showcase some key people and ideas in the domain of theoretical computer science, particularly in computational complexity and algorithms, and related mathematical topics. They show evidence of the considerable scholarship that supports this young field, and they balance an impressive breadth of topics with the depth necessary to reveal the power and the relevance of the work described. Beyond this, the authors discuss the sustained effort of their community, revealing much about the culture of their field. A career in theoretical computer science at the top level is a vocation: the work is hard, and in addition to the obvious requirements such as intellect and training, the vignettes in this book demonstrate the importance of human factors such as personality, instinct, creativity, ambition, tenacity, and luck. The authors' style is characterize d by personal observations, enthusiasm, and humor, and this book will be a source of inspiration and guidance for graduate students and researchers engaged with or planning careers in theoretical computer science.
Proofs
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Author : Source Wikipedia
language : en
Publisher: University-Press.org
Release Date : 2013-09
Proofs written by Source Wikipedia and has been published by University-Press.org this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-09 with categories.
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 60. Chapters: Cantor's diagonal argument, Proof by contradiction, Mathematical induction, Godel's completeness theorem, Mathematical proof, Original proof of Godel's completeness theorem, Q.E.D., Commutative diagram, Conditional proof, Law of large numbers, Turing's proof, Mathematical fallacy, Proof of impossibility, Proof sketch for Godel's first incompleteness theorem, List of published incomplete proofs, Probabilistic proofs of non-probabilistic theorems, Combinatorial proof, Bijective proof, Double counting, Probabilistic method, Structural induction, Probabilistically checkable proof, Infinite descent, Constructive proof, Back-and-forth method, List of mathematical proofs, Elementary proof, Proof without words, Existence theorem, Proof by intimidation, Proof by exhaustion, Direct proof, Proofs from THE BOOK, Proof by contrapositive, Minimal counterexample, Tombstone, Of the form. Excerpt: First published in January 1937 with the title On Computable Numbers, With an Application to the Entscheidungsproblem, Turing's proof was the second proof of the assertion (Alonzo Church proof was first) that some decision problems are "undecidable" there is no single algorithm that infallibly gives a correct YES or NO answer to each instance of the problem. In his own words: ..".what I shall prove is quite different from the well-known results of Godel ... I shall now show that there is no general method which tells whether a given formula U is provable in K ..." (Undecidable p. 145). Turing preceded this proof with two others. The second and third both rely on the first. All rely on his development of type-writer-like "computing machines" that obey a simple set of rules and his subsequent development of a "universal computing machine." In 1905 Jules Richard presented this profound paradox. Alan Turing's first proof constructs this...
The End Of Infinity
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Author : Anthony C. Patton
language : en
Publisher: Algora Publishing
Release Date : 2018-07-01
The End Of Infinity written by Anthony C. Patton and has been published by Algora Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-07-01 with Philosophy categories.
Mathematical Proofs
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Author : Source Wikipedia
language : en
Publisher: Booksllc.Net
Release Date : 2013-09
Mathematical Proofs written by Source Wikipedia and has been published by Booksllc.Net this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-09 with categories.
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 58. Chapters: Back-and-forth method, Bijective proof, Cantor's diagonal argument, Combinatorial proof, Commutative diagram, Conditional proof, Constructive proof, Direct proof, Double counting (proof technique), Elementary proof, Equalization (proof), Law of large numbers, List of incomplete proofs, List of long proofs, List of mathematical proofs, Mathematical fallacy, Mathematical induction, Minimal counterexample, Of the form, Original proof of Godel's completeness theorem, Probabilistically checkable proof, Probabilistic method, Probabilistic proofs of non-probabilistic theorems, Proofs from THE BOOK, Proof by contradiction, Proof by contrapositive, Proof by exhaustion, Proof by infinite descent, Proof by intimidation, Proof of impossibility, Proof sketch for Godel's first incompleteness theorem, Proof without words, Q.E.D., Structural induction, Tombstone (typography), Turing's proof. Excerpt: Turing's proof, is a proof by Alan Turing, first published in January 1937 with the title On Computable Numbers, With an Application to the Entscheidungsproblem. It was the second proof of the assertion (Alonzo Church's proof was first) that some decision problems are "undecidable" there is no single algorithm that infallibly gives a correct YES or NO answer to each instance of the problem. In his own words: ..".what I shall prove is quite different from the well-known results of Godel ... I shall now show that there is no general method which tells whether a given formula U is provable in K ..." (Undecidable p. 145). Turing preceded this proof with two others. The second and third both rely on the first. All rely on his development of type-writer-like "computing machines" that obey a simple set of rules and his subsequent development of a "universal computing machine." In 1905 Jules Richard presented this profound paradox. Alan...
Mathematics And Computation
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Author : Avi Wigderson
language : en
Publisher: Princeton University Press
Release Date : 2019-10-29
Mathematics And Computation written by Avi Wigderson and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-29 with Computers categories.
An introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography
Understanding The Infinite
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Author : Shaughan Lavine
language : en
Publisher: Harvard University Press
Release Date : 2009-06-30
Understanding The Infinite written by Shaughan Lavine and has been published by Harvard University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-30 with Mathematics categories.
An accessible history and philosophical commentary on our notion of infinity. How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge. Praise for Understanding the Infinite “Understanding the Infinite is a remarkable blend of mathematics, modern history, philosophy, and logic, laced with refreshing doses of common sense. It is a potted history of, and a philosophical commentary on, the modern notion of infinity as formalized in axiomatic set theory . . . An amazingly readable [book] given the difficult subject matter. Most of all, it is an eminently sensible book. Anyone who wants to explore the deep issues surrounding the concept of infinity . . . will get a great deal of pleasure from it.” —Ian Stewart, New Scientist “How, in a finite world, does one obtain any knowledge about the infinite? Lavine argues that intuitions about the infinite derive from facts about the finite mathematics of indefinitely large size . . . The issues are delicate, but the writing is crisp and exciting, the arguments original. This book should interest readers whether philosophically, historically, or mathematically inclined, and large parts are within the grasp of the general reader. Highly recommended.” —D. V. Feldman, Choice
A Profile Of Mathematical Logic
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Author : Howard DeLong
language : en
Publisher: Courier Corporation
Release Date : 2012-09-26
A Profile Of Mathematical Logic written by Howard DeLong and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-26 with Mathematics categories.
This introduction to mathematical logic explores philosophical issues and Gödel's Theorem. Its widespread influence extends to the author of Gödel, Escher, Bach, whose Pulitzer Prize–winning book was inspired by this work.
Forever Finite
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Author : Kip K. Sewell
language : en
Publisher: Rond Books
Release Date : 2023-08-01
Forever Finite written by Kip K. Sewell and has been published by Rond Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-01 with Philosophy categories.
INFINITY IS NOT WHAT IT SEEMS… Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes in a divine Creator infinite in knowledge, power, and benevolence. According to this treatise, such assumptions are mistaken. In reality, to be is to be finite. The implications of this assessment are profound: the Universe and even God must necessarily be finite. The author makes a compelling case against infinity, refuting its most prominent advocates. Any defense of the infinite will find it challenging to answer the arguments laid out in this book. But regardless of the reader’s position, Forever Finite offers plenty of thought-provoking material for anyone interested in the subject of infinity from the perspectives of philosophy, mathematics, science, and theology.
The Outer Limits Of Reason
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Author : Noson S. Yanofsky
language : en
Publisher: MIT Press
Release Date : 2016-11-04
The Outer Limits Of Reason written by Noson S. Yanofsky and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-11-04 with Science categories.
This exploration of the scientific limits of knowledge challenges our deep-seated beliefs about our universe, our rationality, and ourselves. “A must-read for anyone studying information science.” —Publishers Weekly, starred review Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own intuitions about the world—including our ideas about space, time, and motion, and the complex relationship between the knower and the known. Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve: • perfectly formed English sentences that make no sense • different levels of infinity • the bizarre world of the quantum • the relevance of relativity theory • the causes of chaos theory • math problems that cannot be solved by normal means • statements that are true but cannot be proven Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.